A) Intensity: [tex]4.8 W/m^2[/tex]
B) Intensity: [tex]2.1 W/m^2[/tex]
C) Total intensity: [tex]13.8 W/m^2[/tex]
D) Distance: [tex]1.38\cdot 10^5 m[/tex]
Explanation:
A)
The relationship between intensity of a sound and power emitted is given by
[tex]I=\frac{P}{A}[/tex]
where
I is the intensity
P is the power
A is the surface area at the distance considered
For the speaker in this problem:
P = 60 W
The distance is r = 1.0 m, so the area to consider is the surface of a sphere with this radius:
[tex]A=4\pi r^2 = 4\pi(1.0)^2=12.6 m^2[/tex]
Therefore, the intensity is
[tex]I=\frac{60}{12.6}=4.8 W/m^2[/tex]
B)
For this problem, we can use the same equation we used before:
[tex]I=\frac{P}{A}[/tex]
where in this case, we have:
P = 60 W is the power of the speaker
r = 1.5 m is the distance considered
Therefore, the area of the spherical surface is
[tex]A=4\pi r^2 = 4\pi(1.5)^2=28.3 m^2[/tex]
And so, the intensity here is
[tex]I=\frac{60}{28.3}=2.1 W/m^2[/tex]
C)
In this problem, we have to consider 4 speakers. In order to find the total intensity, we have to add the intensity of each speakers.
For the two front speakers, we have
[tex]P=60 W[/tex]
r = 1.0 m
So we have already calculated their intensity in part A), and it is
[tex]I_1 = I_2 = 4.8 W/m^2[/tex]
For the two rear speakers, we have
[tex]P=60 W[/tex]
r = 1.5 m
So their intensity has been calculated in part B),
[tex]I_3 = I_4 = 2.1 W/m^2[/tex]
Therefore, the total intensity is
[tex]I=I_1 + I_2 + I_3 + I_4 = 4.8+4.8+2.1+2.1=13.8 W/m^2[/tex]
D)
In this case, we want to find the distance r at which the intensity is
[tex]I=10^{-12} W/m^2[/tex]
Using four speakers with power
P = 0.06 W
Since we have 4 speakers, the intensity due to each speaker at the location considered must be
[tex]I'=\frac{I}{4}=\frac{10^{-12}}{4}=2.5\cdot 10^{-13} W/m^2[/tex]
Using the usual equation, we find the area:
[tex]A=\frac{P}{I'}=\frac{0.06}{2.5\cdot 10^{-13}}=2.4\cdot 10^{11} m^2[/tex]
And so, we can find what is the corresponding distance:
[tex]A=4\pi r^2\\r=\sqrt{\frac{A}{4\pi}}=\sqrt{\frac{2.4\cdot 10^{11}}{4\pi}}=1.38\cdot 10^5 m[/tex]
So, the person must be at 138 km.
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The intensity of sound waves can be computed by using the equation I = P / (4πr²). For the minimum discernible intensity, we employ a rearranged version of the same formula to calculate the distance from the car stereo at the threshold of hearing.
Explanation:The question is about the calculation of sound intensity produced by a car stereo featuring four speakers, each rated at 60 Watts, at various distances. To calculate the intensity (I) of the sound waves at different distances, we use the equation I = P / (4πr²) where P represents power and r is the distance.
For part D, to find the distance (d) at which sound from the stereo could still be discerned, we rearrange the intensity equation to solve for distance, d = √(P / (4πI)). We substitute P with the sound produced (0.06 W) and I with the minimum discernible intensity (10-12 W/m²). Disclaimer: Since external factors like absorption or diffusion are not taken into account, and the car stereo is not treated as a perfect point source emitter, these values serve as approximation.
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A soft-drink bottler purchases glass bottles from a vendor. The bottles are required to have internal pressure strength of at least 150 pounds per square inch (psi). A prospective bottle vendor claims that its production process yields bottles with mean internal pressure strength of 157 psi and standard deviation of 3 psi. The bottler strikes an agreement with the vendor that permits the bottler to sample from the vendor's production process to verify the vendor's claim. The bottler obtains a random sample of 64 bottles. If the mean internal pressure strength of the sample falls below K, the bottler will conclude the vendor's claim about the mean internal pressure strength to be false. Suppose the bottler is willing to risk a 2% chance of concluding the vendor's claim to be false even if the claim is true. Find the value of K.
Answer: K =24 psi
Explanation:
Given: Standard deviation =3psi
Internal pressure strength =157psi
Number of random bottle =n=64
K= 3 × square root of 64
K= 3×8=24 psi
If mean internal pressure K fall below K,
157-1.3=155.7psi
At 2%:
0.16×64 = 10.24
Final answer:
The value of K is 150.835 psi.
Explanation:
To find the value of K, we need to calculate the critical value of the sample mean that separates the lower 2% of the distribution from the upper 98%. First, we need to determine the z-score corresponding to the desired probability of 2%. We can find this value using a standard normal distribution table or a calculator. The z-score for a 2% probability is approximately -2.055. Next, we can calculate the value of K by multiplying the z-score by the standard deviation and adding it to the mean:
K = 157 + (-2.055 * 3) = 150.835 psi
Therefore, if the sample mean falls below 150.835 psi, the bottler will conclude the vendor's claim about the mean internal pressure strength to be false.
A 0.30 kg mass is sliding on a horizontal, frictionless air track with a speed of 4.0 m/s when it instantaneously hits and sticks to a 2.0 kg mass initially at rest on the track. The spring constant is 100 N/m . The other end of the spring is fixed.a. Determine the following for the 0.30 kg mass immediately before the impact:i. Its linear momentum ii. Its KEb. Determine the following for the combined masses immediately after the impact:i. The linear momentum in. The KEc. Besides the fact that the objects stick together, what other clue lets you know that this was an inelastic collision
Answer
given,
mass of block 1, m = 0.3 Kg
speed of block 1, v = 4 m/s
mass of second block,M = 2 Kg
initial speed of block = 0 m/s
spring constant, k = 100 N/m
a) for block 1
linear momentum before collision
P₁ = m v = 0.3 x 4 = 1.2 Kg.m/s
Kinetic energy
[tex]KE_1 = \dfrac{1}{2}mv^2[/tex]
[tex]KE_1 = \dfrac{1}{2}\times 0.3\times 4^2[/tex]
[tex]KE_1 =2.4\ J[/tex]
b) After impact
final velocity calculation
using conservation of momentum
m v = (m + M )v_f
0.3 x 4 = 2.3 x v_f
v_f = 0.522 m/s
Linear momentum
P₂ = (m+M) v_f
P₂ = 1.5 x 0.522
P₂ = 0.783 kg.m/s
Kinetic energy
[tex]KE_2= \dfrac{1}{2}(M+m)v^2[/tex]
[tex]KE_2= \dfrac{1}{2}\times 2.3\times 0.522^2[/tex]
[tex]KE_2=0.313\ J[/tex]
A security guard walks at a steady pace, traveling 120 mm in one trip around the perimeter of a building. It takes him 230 ss to make this trip. what is his speed.
Answer:
0.0005217 m/s or 5.217×10⁻⁴ m/s
Explanation:
Speed: This can be defined as the ratio of the distance covered by a body to the time taken to cover that distance. The S.I unit of speed is m/s. Speed is a scalar quantity, because it can only be represented by magnitude only.
Mathematically, speed is expressed as
S = d/t ......................................................... Equation 1
Where S = speed, d = total distance, t = time taken to cover the distance
Given: d = 120 mm = (120/1000) m = 0.12 m, t = 230 s.
Substituting into equation 1
S = 0.12/230
S = 0.0005217 m/s
Hence the speed of the security guard = 0.0005217 m/s or 5.217×10⁻⁴ m/s
The speed of the security guard is 0.52 mm/ss.
Explanation:To find the speed of the security guard, we need to divide the distance traveled by the time taken. In this case, the distance is 120 mm and the time is 230 ss. The formula to calculate speed is:
Speed = Distance / Time
Substituting the given values:
Speed = 120 mm / 230 ss
Simplifying the expression, we get:
Speed = 0.52 mm/ss
Therefore, the speed of the security guard is 0.52 mm/ss.
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Suppose you are an observer standing on the Moon, looking back at Earth.
If the moon is in the waxingquarter phase, what phase of the Earth would you see?
A. waxing quarter
B. full
C. new
D. waning quarter
E. none; the Earth doesn't have phases
Answer:
D. waning quarter
Explanation:
The Moon is a natural satellite of Earth and reflects the light of Sun to become visible from the Earth. It shows various phases during its revolution around the Earth depending on how much part of the lit up portion of the Moon is towards Earth. Just like Moon shows phases, if we go on Moon the Earth will show various phases.
When the Moon will be in waxing quarter phase i.e. as seen from Earth, we will see a semicircular Moon with its right side lit up. At the same time if someone from the Moon sees Earth, the Earth will show a waning quarter phase. It will be seen as semicircular but left side will be lit up.
If you are an observer standing on the Moon looking back at Earth during the waxing quarter phase of the Moon, you would see a waning quarter phase of the Earth. Thus, the correct option is D.
Explanation:If you are an observer standing on the Moon looking back at Earth, the phase of the Earth that you would see depends on the relative positions of the Moon, Earth, and Sun. In the waxing quarter phase of the Moon, you would see a waning quarter phase of the Earth. This is because when the Moon is in the waxing quarter phase, the illuminated portion of the Moon is on the right side, meaning the Earth would be on the left side of the Moon and in the waning quarter phase.
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When riding a 10-speed bicycle up a hill, a cyclist shifts the chain to a larger-diameter gear attached to the back wheel. Why is this gear preferred to a smaller gear? a. The torque exerted by the chain on the gear is larger. b. The force exerted by the chain on the gear is larger. c. You pedal more frequently to travel the same distance. d. Both a and c are correct.
To solve this problem we will use the concepts of the moment of rotational inertia, angular acceleration and the expression of angular velocity.
The rotational inertia is expressed as follows:
[tex]I = \sum mr^2[/tex]
Here,
m = Mass of the object
r = Distance from the rotational axis
The rotational acceleration in terms of translational acceleration is
[tex]\alpha = \frac{a}{R}[/tex]
Here,
a = Acceleration
R = Radius of the circular path of the object
The expression for the rotational speed of the object is
[tex]\omega = \frac{\Delta \theta}{\Delta t}[/tex]
Here,
[tex]\Delta \theta[/tex] is the angular displacement of the object
The explanation by which when climbing a mountain uphill is changed to a larger pinion, is because it produces a greater torque but it is necessary to make more pedaling to be able to travel the same distance. Basically every turn results in less rotations of the rear wheel. Said energy that was previously used to move the rotation of the wheel is now distributed in more turns of the pedal. Therefore option a and c are correct.
This would indicate that the correct option is D.
Shifting to a larger-diameter gear in a 10-speed bicycle allows for an increased force exerted by the chain on the gear and a greater torque on the wheel, making it easier to ride uphill.
Explanation:When riding a 10-speed bicycle up a hill, shifting to a larger-diameter gear attached to the back wheel is preferred compared to a smaller gear because it increases the force exerted by the chain on the gear and allows the cyclist to exert a greater torque on the wheel.
By shifting to a larger gear, the chain wraps around a larger portion of the gear's circumference, resulting in a greater force being applied to rotate the wheel. This increased force allows the cyclist to overcome gravity more efficiently and climb the hill with less effort.
The larger gear also allows the cyclist to apply a greater torque to the wheel. Torque is the rotational equivalent of force and represents the ability to turn the wheel. With a larger gear, the cyclist can pedal with more force and generate a larger torque, which is necessary to propel the bike up the hill.
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At t=0 a ball, initially at rest, starts to roll down a ramp with constant acceleration. You notice it moves 1 foot between t=0 seconds and t = 1 second. 1)How far does it move between t = 1 second and t = 2 seconds?
Answer: 3 ft
Explanation:
Given:
- the ball is initially at rest before it start moving at constant acceleration.
a = constant
u = initial speed = 0
Using the formula for displacement.
d = ut + 0.5at^2 .....1
Since u = 0, equation 1 becomes;
d = 0.5at^2 .....3
making a the subject of formula
a = 2d/t^2 ....2
Where; a = acceleration, d = distance travelled, t = time taken, u = initial speed
d = 1 ft, t =1 s
Substituting into equation 2 above.
a = (2×1)/1^2 = 2ft/s^2
Since the acceleration is 2ft/s^2, at t= 2 sec the distance it would have covered will be:
Using equation 3;
d = 0.5(2 × 2^2) = 0.5(8) = 4ft
therefore, the distance travelled between t = 1 to t= 2
Is :
d2 = d - d1
Where d = total distance (t=0 to t= 2) =4 ft
d1 = distance covered between (t=0 to t=1) = 1 ft
d2 = 4 - 1 = 3ft
The distance traveled between time 1s and time 2s is 3 ft.
The given parameters:Initial velocity of the ball, u = 0Acceleration = constantThe distance traveled by the ball at time t is calculated as follows;
[tex]s = ut + \frac{1}{2} at^2\\\\s = 0 + \frac{1}{2} at^2\\\\s = \frac{1}{2} at^2\\\\a = \frac{2s}{t^2}[/tex]
when the distance, s = 1 ft and time, t = 1 s
[tex]a = \frac{2(1)}{(1)^2} \\\\a = 2 \ ft/s^2[/tex]
When the time of motion, t = 2s, the distance traveled is calculated as;
[tex]s = \frac{1}{2} (2) (2)^2\\\\s = 4 \ ft[/tex]
The distance traveled between t =1 and t = 2 is calculated as follows;
[tex]s = s_2 - s_1\\\\s = 4 \ ft - 1\ ft\\\\s = 3 \ ft[/tex]
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Find a unit vector that has the same direction as the given vector. −3i + 2j − k
The unit vector that has the same direction as the given vector [tex]\(-3i + 2j - k\) is \(\frac{-3}{\sqrt{14}}i + \frac{2}{\sqrt{14}}j - \frac{1}{\sqrt{14}}k\).[/tex]
To find a unit vector in the same direction as the given vector [tex]\(-3i + 2j - k\)[/tex], we first need to calculate the magnitude of the given vector and then divide each component by that magnitude.
The magnitude of the given vector is given by the formula:
[tex]\[|v| = \sqrt{(-3)^2 + (2)^2 + (-1)^2} = \sqrt{14}\][/tex]
Now, we can find the unit vector [tex]\(\hat{u}\)[/tex] by dividing each component of the given vector by its magnitude:
[tex]\[\hat{u} = \frac{-3}{\sqrt{14}}i + \frac{2}{\sqrt{14}}j - \frac{1}{\sqrt{14}}k\]\[\hat{u} = \frac{-3}{\sqrt{14}}i + \frac{2}{\sqrt{14}}j - \frac{1}{\sqrt{14}}k\][/tex]
So, the unit vector that has the same direction as the given vector [tex]\(-3i + 2j - k\) is \(\frac{-3}{\sqrt{14}}i + \frac{2}{\sqrt{14}}j - \frac{1}{\sqrt{14}}k\).[/tex]
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To find a unit vector with the same direction as the given vector -3i + 2j - k, divide the given vector by its magnitude.
Explanation:To find a unit vector with the same direction as the given vector -3i + 2j - k, we need to divide the given vector by its magnitude. The magnitude of the given vector is the square root of the sum of the squares of its components, which is sqrt((-3)^2 + 2^2 + (-1)^2) = sqrt(14).
So, the unit vector with the same direction as the given vector is (-3i + 2j - k) / sqrt(14).
Particle A of charge 3.06 10-4 C is at the origin, particle B of charge -5.70 10-4 C is at (4.00 m, 0), and particle C of charge 1.08 10-4 C is at (0, 3.00 m). We wish to find the net electric force on C.
Answer:
F_net = 26.512 N
Explanation:
Given:
Q_a = 3.06 * 10^(-4 ) C
Q_b = -5.7 * 10^(-4 ) C
Q_c = 1.08 * 10^(-4 ) C
R_ac = 3 m
R_bc = sqrt (3^2 + 4^2) = 5m
k = 8.99 * 10^9
Coulomb's Law:
F_i = k * Q_i * Q_j / R_ij^2
Compute F_ac and F_bc :
F_ac = k * Q_a * Q_c / R^2_ac
F_ac = 8.99 * 10^9* ( 3.06 * 10^(-4 ))* (1.08 * 10^(-4 )) / 3^2
F_ac = 33.01128 N
F_bc = k * Q_b * Q_c / R^2_bc
F_bc = 8.99 * 10^9* ( 5.7 * 10^(-4 ))* (1.08 * 10^(-4 )) / 5^2
F_bc = - 22.137 N
Angle a is subtended between F_bc and y axis @ C
cos(a) = 3 / 5
sin (a) = 4 / 5
Compute F_net:
F_net = sqrt (F_x ^2 + F_y ^2)
F_x = sum of forces in x direction:
F_x = F_bc*sin(a) = 22.137*(4/5) = 17.71 N
F_y = sum of forces in y direction:
F_y = - F_bc*cos(a) + F_ac = - 22.137*(3/5) + 33.01128 = 19.72908 N
F_net = sqrt (17.71 ^2 + 19.72908 ^2) = 26.5119 N
Answer: F_net = 26.512 N
Final answer:
The question requires applying Coulomb's law to calculate the net electric force on particle C due to particles A and B, and summing the vector forces to find the resultant.
Explanation:
The question involves finding the net electric force on particle C, which is at the position (0, 3.00 m) in the presence of particles A and B with given charges and positions. The situation can be analyzed using Coulomb's law, which states that the force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them.
We would sum the vector forces from both charges on C to find the resultant force. This is a classical physics problem typically addressed in high school or introductory college physics courses.
A horse and a squirrel participate in a race over a 1.20 km long course. The horse travels at a speed of 19.0 m/s and the squirrel can do 3.50 m/s. The horse runs for 0.960 km and then stops to tease the slow-moving squirrel, which eventually passes by. The horse waits for a while after the squirrel passes and then runs toward the finish line. Both animals cross the finish line at the exact same instant. Assume both animals, when moving, move steadily at their respective speeds.
(a)
How far (in m) is the squirrel from the finish line when the horse resumes the race?
m
(b)
For how long in time (in s) was the horse stationary?
s
Answer:
44.21053 m
279.69925 seconds
Explanation:
Time taken by the squirrel to reach the finish line
[tex]\dfrac{1200}{3.5}=342.85714\ seconds[/tex]
Time taken by the horse to cover 0.96 km
[tex]\dfrac{960}{19}=50.52631\ seconds[/tex]
Time taken by the horse to cover 1.2 km
[tex]\dfrac{1200}{19}=63.15789\ seconds[/tex]
Let the distance that the squirrel is from the finish line when the horse resumes the race be x.
During this time the horse also reaches the finish line.
We deduce that
The time taken by the horse from one stop = Time taken by squirrel before x distance from finish line.
[tex]3.5\times (63.15789-50.52631)=x\\\Rightarrow x=44.21053\ m[/tex]
The squirrel is 44.21053 m from the finish line when the horse resumes the race.
Duration is given by
[tex](342.85714-63.15789)=279.69925\ s[/tex]
The duration is 279.69925 seconds
When the horse resumed the race, the squirrel was 240m from the finish line. The horse was stationary for approximately 68.57 seconds.
Explanation:The subject of this question is Physics, specifically it deals with the concept of speed and distance. The grade level is high school as it involves basic kinematics.
(a) The horse has traveled 0.96 km, which is 960 m. It leaves 240 m for the rest of the journey. When the horse stopped, the squirrel kept moving. By the time the horse started again, the squirrel must have traveled more than 240 m. We know the horse and the squirrel finished the race at the same time, hence, they must have started the remaining journey at the same time. So, the distance from the finish line where the squirrel was when the horse began to run again equals to the rest of the horse's journey, which is 240m.
(b) To figure out how long the horse was stationary we need to calculate how long it took the squirrel to cover the distance. This is done by dividing the distance the squirrel traveled (240m) by its speed (3.5 m/s). Hence, time = distance / speed = 240m / 3.5 m/s = 68.57 seconds.
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One planet is three times farther from the Sun than another. Will the farther planet take more, less, or the same amount of time to orbit the Sun
The planet will take more time to orbit the Sun.
Explanation:
According to the Kepler's law of orbital motion, a planet which is far away from the Sun experiences a lower gravitational pull towards the Sun, thus it will move with a lower speed in its orbit. Thus the farther planet takes more time to orbit the Sun compared to the planets closer to the Sun.
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A box is moving along the x-axis and its position varies in time according to the expression:
x = (6.00 m/s^2)t^2,
where x is in meters and t is in seconds.
(a) Determine its position (in m) at t = 3.20 s.
(b) Determine its position (in m) at t = (3.20 s) + Δt.
(c) Evaluate the limit of Δx Δt as Δt approaches zero to find the velocity (in m/s) at t = 3.20 s.
Answer:
38.4 m/s
Explanation:
a) at t = 3.2s. [tex]x = 6 * 3.2^2 = 61.44 m[/tex]
b) at t = 3.2 + Δt. [tex]x = 6*(3.2 + \Delta t)^2[/tex]
c) As Δt approaches 0. We can find the velocity by the ratio of Δx/Δt
[tex]v = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{\Delta t}[/tex]
[tex]v = \frac{6*(3.2 + \Delta t)^2 - 61.44}{\Delta t}[/tex]
[tex]v = \frac{6(3.2^2 + 6.4\Delta t + \Delta t^2) - 61.44}{\Delta t}[/tex]
[tex]v = \frac{61.44 + 38.4\Delta t + \Delta t^2 - 61.44}{\Delta t}[/tex]
[tex]v = \frac{\Delta t(38.4 + \Delta t)}{\Delta t}[/tex]
[tex] v = 38.4 + \Delta t[/tex]
As Δt approaches 0, v = 38.4 + 0 = 38.4 m/s
a. The position of box at t = 3.20s is 61.44 meters
b. The position at t = 3.20+ Δt is, [tex]x(3.20+ \Delta t)=6*(3.20+ \Delta t)^{2}[/tex]
c. The velocity is 38.4 meter per second.
The position of box is given by as a function shown below,
[tex]x(t)=6t^{2}[/tex]
where x is in meters and t is in seconds.
a. The position of box at t = 3.20s is,
[tex]x(3.2)=6*(3.2)^{2}\\ \\x(3.2)=61.44m[/tex]
b. The position at t = 3.20+ Δt,
[tex]x(3.20+ \Delta t)=6*(3.20+ \Delta t)^{2}[/tex]
c. We have to find
[tex]\frac{\Delta x}{\Delta t}=\frac{x(3.20+\Delta t)-x(3.2)}{\Delta t}\\\\\frac{\Delta x}{\Delta t}=\frac{6[(3.2)^{2}+(\Delta t)^{2}+6.4\Delta t - (3.2)^{2} ]}{\Delta t} \\\\\frac{\Delta x}{\Delta t}=6 \Delta t +38.4[/tex]
When [tex]\Delta t[/tex] approaches to zero.
Velocity = [tex]\frac{\Delta x}{\Delta t}=38.4 m/s[/tex]
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Select the options that best complete the statement.
Positively charged particle trajectories
(always, never, the same as)
follow electric field lines, because
(electric field lines are defined by the path positive test charges travel.,
the particle velocities may or may not be in the same direction as the electric field lines.,
positive charges repel each other and, therefore, are repelled by electric field lines.,
the electric force on a positively charged particle is in the same direction as the electric field.)
Positively charged particles always follow the path set by electric field lines. This is because electric field lines are defined by the path that the positive test charges would naturally travel.
Explanation:The options that best complete the statement are 'always' and 'electric field lines are defined by the path positive test charges travel.' This is because the trajectory of positively charged particle trajectories always follow the direction of electric field lines. The electric field lines are essentially mapped out pathways that positive test charges would naturally follow due to the electric force acting upon them. Hence, a positively charged particle will always move along these field lines, from a region of higher potential to a region of lower potential.
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A beam of monochromatic light is diffracted by a slit of width 0.580 mm. The diffraction pattern forms on a wall 1.56 m beyond the slit. The width of the central maximum is 2.05 mm.
Calculate the wavelength of the light.
Answer:
λ = 381 nm
Explanation:
given,
width of slit,a = 0.580 x 10⁻³ m
distance of the screen from the slit, D = 1.56 m
width of central maximum,y = 2.05 x 10⁻³ m
Distance of the edge of central maximum from center of the central maximum y ‘ = y/ 2
Y ‘ = 1.025 x 10⁻³ m
for first maximum
d sin θ = λ
sin θ = tan θ = y'/D
[tex]\lambda = \dfrac{y'a}{D}[/tex]
[tex]\lambda = \dfrac{1.025\times 10^{-3}\times 0.580\times 10^{-3}}{1.56}[/tex]
λ = 381 x 10⁻⁹ m
λ = 381 nm
wavelength of light is equal to 381 nm.
3. Find the de Broglie wavelength of the following particles: (i) An electron in a semiconductor having average thermal velocity at T = 300 K
Answer:
The wavelength of the electron is [tex]\dfrac{4.4097\times10^{-9}}{\sqrt{a}}\ m[/tex]
Explanation:
Given that,
Temperature = 300 K
We know that,
The energy of free electron is
[tex]E=\dfrac{(\hbar)^2k^2}{2m}[/tex]
[tex]k=\dfrac{\sqrt{2mE}}{\hbar}[/tex]
Where, k = wave number
The momentum of the electron is
[tex]p=\hbar k[/tex]
Th effective mass is
[tex]m=am_{0}[/tex]
We need to calculate the wavelength of the electron
Using formula of wave number
[tex]k=\dfrac{2\pi}{\lambda}[/tex]
[tex]\lambda=\dfrac{2\pi}{k}[/tex]
Put the value of k
[tex]\lambda=\dfrac{2\pi}{\dfrac{\sqrt{2mE}}{\hbar}}[/tex]
[tex]\lambda=\dfrac{h}{\sqrt{2am_{0}E}}[/tex]
We know that,
Thermal energy of electron
[tex]E=3kT[/tex]
The de Broglie wavelength of the electron is
[tex]\lambda=\dfrac{h}{\sqrt{2am_{0}\times3kT}}[/tex]
Put the value into the formula
[tex]\lambda=\dfrac{6.63\times10^{-34}}{\sqrt{2\times9.1\times10^{-31}\times3\times300\times1.380\times10^{-23}\times a}}[/tex]
[tex]\lambda=\dfrac{4.4097\times10^{-9}}{\sqrt{a}}\ m[/tex]
Hence, The wavelength of the electron is [tex]\dfrac{4.4097\times10^{-9}}{\sqrt{a}}\ m[/tex]
The de Broglie wavelength of an electron can be calculated using de Broglie's equation, λ = h/p. To find the wavelength for an electron in a semiconductor at 300 K, one needs to know the electron's velocity, which is related to its kinetic energy at that temperature. Precise calculation would require details about the effective electron mass in the semiconductor.
Explanation:The de Broglie wavelength λ of a particle can be determined using de Broglie's equation, λ = h/p, where 'h' is Planck's constant (6.626 × 10-34 m2 kg/s) and 'p' is the momentum of the particle. The momentum of an electron moving at thermal velocities in a semiconductor at room temperature (T = 300 K) can be approximated using the formula p = mv, where 'm' is the mass of an electron (9.11 × 10-31 kg) and 'v' is the velocity. The average thermal velocity can be obtained from the kinetic theory of gases, which states that the average kinetic energy (KE) of a particle is ⅓kT, where 'k' is the Boltzmann constant (1.38 × 10-23 J/K) and 'T' is the temperature in kelvins. Therefore, v can be calculated as √(3kT/m). Substituting this velocity into the momentum formula and then into de Broglie's equation yields the de Broglie wavelength of the electron.
However, a precise value for the de Broglie wavelength of an electron in a semiconductor at T = 300 K would require additional information not provided in the question, such as the effective mass of the electron in the semiconductor, which can differ from the free electron mass due to the crystal structure's influence.
In a salt crystal, the distance between adjacent sodium and chloride ions is 2.82 Angstroms. What is the force of attraction between the two singly charged ions
The force of attraction between sodium and chloride ions in a salt crystal can be calculated using Coulomb's law, considering isotropic forces and applying the Madelung constant.
Explanation:
The force of attraction between the sodium and chloride ions in a salt crystal can be calculated using Coulomb's law, which involves the product of the charges divided by the distance between them squared. The force is proportional to the ionic charges and inversely proportional to the square of the distance between the ions. In a sodium chloride (NaCl) crystal, the resulting ions (Na+ and Cl¯) produce isotropic forces, which means the force is the same in all directions.
Given the distance mentioned in your question (2.82 Angstroms), you can use this along with the known charges of sodium and chloride ions to calculate the force. However, please note that this calculation needs to take into account the Madelung constant, which is a factor for considering the interaction of a sodium ion with all the nearby chloride and sodium ions in a tightly arranged three-dimensional lattice structure. For a NaCl crystal, the Madelung constant is approximately 1.75.
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An undamped 2.65 kg horizontal spring oscillator has a spring constant of 38.5 N/m. While oscillating, it is found to have a speed of 2.92 m/s as it passes through its equilibrium position. What is its amplitude of oscillation?
Answer: 0.44831m
Explanation:
Unwanted horizontal spring oscillator=2.65kg
Spring constant =38.5N/M
Speed=2.92m/s
Amplitude of oscillation=?
Potential energy=m*v2/2
=2.65*2.92/2
=3.8695J
Potential energy=kinetic energy
Potential energy=1/2kx^2
3.869=1/2*38.5*x^2
3.869=19.25x^2
Dividing both sides by 19.25
3.869/19.25=x^2
So therefore, x^2=√0.200987
x=0.44831m
Two cars, one of mass 1300 kg, and the second of mass 2400 kg, are moving at right angles to each other when they collide and stick together. The initial velocity of the first car is 12 m/s in the positive x direction and that of the second car is 18 m/s in the positive y direction.
What is the magnitude of the velocity of the wreckage of the two cars immediately after the collision?
To solve this problem we will apply the momentum conservation theorem, that is, the initial momentum of the bodies must be the same final momentum of the bodies. The value that will be obtained will be a vector value of the final speed of which the magnitude will be found later. Our values are given as,
[tex]m_1 = 1300kg[/tex]
[tex]m_2 = 2400kg[/tex]
[tex]u_1 = 12m/s i[/tex]
[tex]u_2 = 18m/s j[/tex]
Using conservation of momentum,
[tex]m_1u_1+m_2u_2 = (m_1+m_2)v_f[/tex]
[tex]1300*12i-2400*18j = (1300+2400)v_f[/tex]
Solving for [tex]v_f[/tex]
[tex]v_f = 4.2162i-11.6756j[/tex]
Using the properties of vectors to find the magnitude we have,
[tex]|v| = \sqrt{(4.2162^2)+(-11.6756)^2}[/tex]
[tex]|v| = 12.4135m/s[/tex]
Therefore the magnitude of the velocity of the wreckage of the two cars immediately after the collision is 12.4135m/s
For the two cars the final velocity of the wreckage of the two cars immediately after the collision is 12.4135 m/s.
What is conservation of momentum?Momentum of a object is the force of speed of it in motion. Momentum of a moving body is the product of mass times velocity.
When the two objects collides, then the initial collision of the two body is equal to the final collision of two bodies by the law of conservation of momentum.
The mass of the car one is 1300 kg and the mass of the second car is 2400 kg.
The initial velocity of the first car is 12 m/s in the positive x direction, and that of the second car is 18 m/s in the positive y direction.
Final velocity of the two cars after the collision is equal. Thus, the final velocity of the wreckage of the two cars immediately after the collision by the law of conservation can be given as,
[tex](1300)12\hat i-(2400)18\hat j=(1300+2400)v\\v=4.2162\hat i-11.6756\hat j[/tex]
Solve the above equation further using the property of vectors as,
[tex]v=\sqrt{(4.2162)^2+(-11.6756)^2}\\v=12.4135\rm m/s[/tex]
Hence, For the two cars the final velocity of the wreckage of the two cars immediately after the collision is 12.4135 m/s.
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A cube with sides of area 32 cm2 contains a 35.9 nanoCoulomb charge. Find the flux of the electric field through the surface of the cube in unis of Nm2/C.
Answer:
The electric flux through the surface is equal to 3.878 x 10³ Nm²/C
The field distance r is equal to half the length of each side of the cube.
From the area the length of each size was calculated and the field distance and charge were used in calculating the magnitude of the electric field vector which was found to be 202 x 10³ N/C
The total flux area available to this electric field is 6x32cm²
Explanation:
The full solution can be found in the attachment below.
Thank you for reading this post and I hope it is helpful to you.
One simple model for a person running the 100m dash is to assume the sprinter runs with constant acceleration until reaching top speed, then maintains that speed through the finish line.
If a sprinter reaches his top speed of 11.3m/s in 2.44s , what will be his total time?
Answer:
Explanation:
Given
Race length [tex]L=100\ m[/tex]
Runner reaches maximum velocity in [tex]t_1=2.4\ s[/tex]
[tex]v_{max}=11.3\ m/s[/tex]
distance traveled during this time
[tex]v=u+at[/tex]
[tex]11.3=0+a\times 2.4[/tex]
[tex]a=\frac{11.3}{2.4}=4.708\ m/s^2[/tex]
distance traveled in this time
[tex]s=ut+\frac{1}{2}at^2[/tex]
[tex]s=0+0.5\times 4.708\times (2.44)^2[/tex]
[tex]s=14.014\ m[/tex]
Remaining distance will be traveled with [tex]v_{max}[/tex]
remaining distance [tex]d=100-14.014=85.98\ m[/tex]
time taken [tex]t_2=\frac{d}{v_{max}}=\frac{85.98}{11.3}=7.609\ s[/tex]
total time [tex]t=t_1+t_2[/tex]
[tex]t=2.44+7.609=10.049\ s[/tex]
Assuming a Chevy Cruze may be operated indefinitely at 30 m/s with no stops for provisions,
and may be refueled in transit, how long (in years) would be required for the Chevy
vehicle to travel one light year?
A. 1,000 years
B. 100,000 years
C. 1,000,000 years
D. 10,000,000 years
To solve this problem we will apply the linear motion kinematic equations. In which the speed is defined as the distance traveled in a certain period of time. In turn we must emphasize that 1 light year is equivalent to [tex]9.461*10^{15}m[/tex]. Mathematically the speed is described as,
[tex]v = \frac{x}{t} \rightarrow t = \frac{x}{v}[/tex]
Replacing,
[tex]t = \frac{9.461*10^{15}}{30}[/tex]
[tex]t = 3.15366*10^{14}s (\frac{1 year}{3.154*10^7})[/tex]
[tex]t \approx 10,000,000 years[/tex]
Therefore the correct answer is D.
Suppose that a single guitar player makes noise at 70 dB. (a) If two guitar players play at the same time, what is the noise level in dB? What is the sound intensity in W/m2 ?
Answer:
Explanation:
Given
For single guitar noise level [tex]SL=70\ dB[/tex]
Intensity of sound
Sound level[tex]=10\log (\frac{I}{I_0})[/tex]
where I=Intensity of sound Produced
[tex]I_0=[/tex]Human threshold frequency [tex](10^{-12}\ W/m^2)[/tex]
[tex]70=10\log (\frac{I}{10^{-12}})[/tex]
[tex]I=10^{-12}\times 10^7[/tex]
[tex]I=10^{-5}\ W/m^2[/tex]
For 2 guitars
[tex]I'=2I=2\times 10^{-5}\ W/m^2[/tex]
[tex]SL=10\log (\frac{I}{I_0})[/tex]
[tex]SL=10\log (\frac{2\times 10^{-5}}{10^{-12}})[/tex]
[tex]SL=10(7+\log (2))[/tex]
[tex]SL=73.01\ dB[/tex]
Is it possible for a car to be accelerating to the west while it is driving to the east?
Answer:
Yes
Explanation:
If the acceleration has an opposite direction to the velocity of the car, this means that it is opposed to is motion. Therefore, it is called deceleration, since the car's velocity will decrease until it stops and then will start it moving towards the west.
Protons and neutrons are made from combinations of the two most common quarks, the u quark and the d quark. The u quark's charge is +2/3e, while the d quark carries −1/3e.
Part A
How could three of these quarks combine to make a proton?
A. uuu
B. uud
C. udd
D. ddd
Part B
How could three of these quarks combine to make a neutron?
A. uuu
B. uud
C. udd
D. ddd
Answer:
Part A B. Part B C.
Explanation:
A) The elementary charge e (without sign) is equal to the charge of one electron (with negative sign) or to the charge of one proton (with positive sign), so the proton must have a total charge of +e.
if u = +2/3 e and d= -1/3 e, we need a combination which sum gives +3/3 e.
Combination A adds to 6/3e, so it is not possible. C adds to zero, and D gives a negative result.
The only remaining choice is udd:
uud ⇒ +2/3 e + 2/3 e -1/3 e = +3/3 e = +e
So, the statement B is true.
B) As the neutron has no net charge, we need to find a combination which sum adds to zero.
So, A and D are not possible, as they are combinations of the same type of quarks, so the sum is either positive or negative, but not zero.
uud gives +e (we chose it to make a proton in part A), so the only remaining choice is udd:
udd⇒ +2/3 e -1/3 e - 1/3 e = 0
So the statement C is true.
A sprinter in a 100-m race accelerates uniformly for the first 71 m and then runs with constant velocity. The sprinter’s time for the first 71 m is 9.9 s. Determine his acceleration.
Answer:
The acceleration of the sprinter is 1.4 m/s²
Explanation:
Hi there!
The equation of position of the sprinter is the following:
x = x0 + v0 · t + 1/2 · a · t²
Where:
x = position of the sprinter at a time t.
x0 = initial position.
v0 = initial velocity.
t = time.
a = acceleration.
Since the origin of the frame of reference is located at the starting point and the sprinter starts from rest, then, x0 and v0 are equal to zero:
x = 1/2 · a · t²
At t = 9.9 s, x = 71 m
71 m = 1/2 · a · (9.9 s)²
2 · 71 m / (9.9 s)² = a
a = 1.4 m/s²
The acceleration of the sprinter is 1.4 m/s²
The sprinter's acceleration can calculated by solving two equations from physics: the first formula of motion (final velocity equals initial velocity plus acceleration times time) and the formula for final velocity (distance divided by time). The result is an approximate acceleration of 0.72 m/s^2.
Explanation:The question requires the use of kinematics, a topic in physics. The formula to calculate uniform acceleration is given by a = 2*(final velocity - initial velocity) / time. Here, final velocity is the velocity achieved after 71m, initial velocity is 0 (standing start), and time is 9.9s. However, we are not given final velocity. We will have to use the first formula of motion, v = u + at, in which v is the final velocity, u is the initial velocity, a is acceleration and t is time. Simultaneously, we have v = d/t where d is distance and t is time. Solving these two equations will give the acceleration (a).
From the first formula of motion and the equation v = d/t, it's clear the sprinter's final velocity (when he/she stops accelerating) is 71m/9.9s which equals approximately 7.17 m/s. Substituting into the first formula of motion gives 7.17 m/s = 0 + a*9.9s which simplifies to a = 7.17 m/s / 9.9s. This gives us an acceleration of approximately 0.72 m/s2.
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I have a bag that contains 976 mL of air at room temperature (25 oC) and has a pressure of 795 torr. If I ascend a mountain and the bag does not break, what is the volume in liters (L) when the pressure is 553 torr and the temperature is 10. oC?
Final answer:
The volume of the bag at the new pressure and temperature is 0.991 L.
Explanation:
To solve this problem, we can use the combined gas law:
P1V1/T1 = P2V2/T2
Where:
P1 = 795 torr (initial pressure)
V1 = 976 mL (initial volume, convert to L: 976 mL / 1000 = 0.976 L)
T1 = 25°C (initial temperature, convert to Kelvin: 25°C + 273 = 298 K)
P2 = 553 torr (final pressure)
T2 = 10°C (final temperature, convert to Kelvin: 10°C + 273 = 283 K)
Now we can plug in these values into the equation:
P1V1/T1 = P2V2/T2
(795 torr)(0.976 L)/(298 K) = (553 torr)(V2)/(283 K)
Solving for V2, we get:
V2 = (795 torr)(0.976 L)(283 K) / (553 torr)(298 K) = 0.991 L
Therefore, the volume of the bag at the new pressure and temperature is approximately 0.991 L.
A rope is lying on the floor and has a mass which applies a force of 3 N/m. How much work is required to raise one end of the rope to a height of 2 meters?
Answer:
At least 6 N, assuming that the rope has a length of more than 2 meters.
Explanation:
In general, if it takes a force of [tex]F(h)[/tex] newtons to lift an object at height [tex]h[/tex], the work done lifting the object from [tex]h = a[/tex] to [tex]h = b[/tex] can be found using the definite integral about [tex]h[/tex]:
[tex]\displaystyle W = \int \limits_{a}^{b} F(h)\, dh[/tex].
If the value of [tex]F(h)[/tex] is a constant [tex]m \cdot g[/tex] regardless of height [tex]h[/tex], then the result of the integral would be
[tex]\displaystyle \int \limits_{a}^{b} (m \cdot g)\, dh = \left[m \, g\, h \right]_a^b = m\cdot g \, (b - a)[/tex].
However, in this case the value of [tex]F(h)[/tex] does depend on the value of [tex]h[/tex].
At height [tex]h = 0\; \rm m[/tex], nothing is being lifted. The amount of force required would be zero. At height [tex]h = 1\; \rm m[/tex], one meter of the rope is in the air. That requires a force of at least [tex]1\; \rm m \times 3\; N \cdot m^{-1} = 3\; N[/tex].In general, at a height of [tex]h[/tex] meters, the force required would be at least [tex]3\, h[/tex] Newtons.In other words, [tex]F(h) = 3\; h[/tex] where [tex]F[/tex] is in Newtons and [tex]h[/tex] is in meters.
Evaluate the integral:
[tex]\begin{aligned} W &= \int \limits_{a}^{b} F(h)\, dh \cr &= \int \limits_{0}^{2}3\, h \, dh && \text{Apply the power rule.}\cr &= \left[\frac{3}{2}\,h^2\right]_{0}^{2}\cr &= \frac{3}{2} \times 2^2 \cr &= 6\end{aligned}[/tex].
The work required to raise one end of the rope to a height of 2 meters with an applied force of 3 N is 6 joules.
We can calculate the work required to raise one end of the rope to a height of 2 meters as follows:
[tex] W = F*d*cos(\theta) [/tex]
Where:
F: is the force exerted = 3 N (newton: unit of force)
d: is the displacement = 2 m
θ: is the angle between the applied force and the displacement
Since the force and the displacement are in the same direction, θ = 0, so:
[tex] W = F*d*cos(\theta) = Fdcos(0) = F*d [/tex]
Hence, the work done is:
[tex] W = 3 N*2m = 6 J [/tex]
Therefore, it is required 6 J of work.
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How far apart must two point charges of 75.0 nC (typical of static electricity) be to have a force of 1.00 N between them?
Answer:
0.71 cm
Explanation:
[tex]q_1=75.0nC=75\times 10^{-9}C[/tex]
[tex]1nC=10^{-9}C[/tex]
[tex]q_2=75.0nC=75\times 10^{-9}C[/tex]
Force between two charges=1 N
Coulomb's law of force
[tex]F=\frac{kq_1q_2}{r^2}[/tex]
Where k=[tex]9\times 10^9Nm^2/C^2[/tex]
Using the formula
[tex]1=\frac{9\times 10^9\times 75\times 10^{-9}\times 75\times 10^{-9}}{r^2}[/tex]
[tex]r^2=0.50625\times 10^{-4}[/tex]m
[tex]r=\sqrt{0.50625\times 10^{-4}}=0.71\times 10^{-2} m[/tex]
[tex]r=0.71\times 10^{-2}\times 10^{2}=0.71\times 10^{-2+2}=0.71\times 10^0=0.71 cm[/tex]
Using formula
[tex]1 m=10^2cm, a^x\cdot a^y=a^{x+y}, a^0=1[/tex]
Hence, the distance between two charges =r=0.71 cm
A proton is released in a uniform electric field, and it experiences an electric force of 2.04×10−14 N toward the south.
What is the magnitude of the electric field?
What is the direction of the electric field?
a. toward the north
b.toward the south
c. toward the east
d. toward the west
Answer:
Part A:
[tex]E=127500N/C\\E=1.275*10^{5} N/C[/tex]
Part B:
Option B (Towards the South)
Explanation:
Part A:
Magnitude if electric field E:
E=Force/charge
Force=2.04×10−14 N
Charge=1.6×10−19 C
[tex]E=\frac{2.04*10^{-14}}{1.6*10^{-19}} \\E=127500N/C\\E=1.275*10^{5} N/C[/tex]
Part B:
Option B (Towards the South)
As electron is experiencing the force towards south,it means the direction of the electric field is towards the south because direction of field lines is from positive to negative, so proton is moving towards south it means negative charge is in south to which proton is attracted. So electric field is towards South.
Final answer:
The correct option is (b) toward the south. The magnitude of the electric field is 1.275 N/C, and the direction of the electric field is toward the south, as it attracts the positively charged proton in that direction.
Explanation:
To determine the magnitude of the electric field experienced by a proton, we use the formula for electric force: F = Eq, where F is the electric force, E is the electric field strength, and q is the charge of the particle.
Given that the force experienced by the proton is 2.04×10⁻¹⁴ N and the charge of a proton (q) is approximately 1.6×10⁻¹⁹ C, we can rearrange the formula to solve for E: E = F/q. Substituting the values, we obtain E = (2.04×10⁻¹⁴ N) / (1.6×10⁻¹⁹ C) = 1.275 N/C. Therefore, the magnitude of the electric field is 1.275 N/C.
Regarding the direction of the electric field, it is crucial to understand that electric fields exert forces on positive charges away from the source of the field and toward negative charges or areas of lower potential.
Since the proton is positively charged and experiences a force toward the south, the electric field's direction must be toward the south to attract the proton in that direction. So, the correct choice for the direction of the electric field is (b) toward the south.
When a second student joins the first, the height difference between the liquid levels in the right and left pistons is 40 cm . What is the second student's mass?
To find the second student's mass, we can use the principle of Pascal's Law which states that when a fluid is at rest in a container, the pressure is transmitted equally in all directions. The height difference between the liquid levels in the right and left pistons can be used to determine the mass.
Explanation:To find the second student's mass, we can use the principle of Pascal's Law which states that when a fluid is at rest in a container, the pressure is transmitted equally in all directions. In this case, the height difference between the liquid levels in the right and left pistons can be used to determine the mass. The height difference is 40 cm, so the pressure difference is equal to the weight of the second student:
P = ρgh, where P is the pressure difference, ρ is the density of the liquid, g is the acceleration due to gravity, and h is the height difference.
Using the formula, we can solve for the mass of the second student:
m = P/(g*h), where m is the mass of the second student.
Let's plug in the values:
m = (ρgh)/(g*h) = ρ. So, the mass of the second student is equal to the density of the liquid.
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To find the mass of the second student using the difference in liquid levels in a hydraulic system, we can rearrange the pressure difference equation, which takes into account the height difference, gravity, and fluid density.
Explanation:The subject of the question delves into the physical principles of hydraulics. If the height difference between the liquid levels in the right and left pistons is 40 cm when a second student joins the first, we can apply the principle of hydraulic machines which is essentially the conservation of energy principle.
In hydraulics, pressure applied on one piston gets transmitted to the other without being diminished. Hence, in this context, let's denote the mass of the second student as m2. The difference in heights or pressure difference ΔP can be measured by subtracting fluid devices and can be expressed using the formula ΔP = m2g
In this equation, 'g' is the acceleration due to gravity. Given the height difference Δh as 40cm, we can deduce the density (ρ) of the fluid. Remember, pressure difference ΔP = ρg Δh. Hence, we can obtain the mass of the second student by rearranging the equation as follows: m2 = (ρg Δh) / g. Please note: for this scenario, the value of 'g' needs to be considered as 9.8 m/s² (earth's acceleration) and ρ is the fluid density (e.g., for water, it is 1000 kg/m³).
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An electromagnetic wave in a vacuum traveling in the +x direction generated by a variable source initially has a
wavelength ? of 225 ?mand a maximum electric field Emax in the +ydirection of 7.40�10^-3 V/m .
If the period of the wave is then increased by a factor of 1.50, what is the equation of the
resulting magnetic field component of the wave?
What are the coefficients in the equation for the magnetic field component
of the wave after the period increases by a factor of 1.50?
Enter the maximum magnetic field, wavenumber,
and angular frequency in teslas, inverse meters, and radians per second separated by commas.
Answer:
B (z) = 2,467 10⁻¹¹ cos (1,486 10⁸ x - 5.571 10¹⁵ t )
Explanation:
An electromagnetic wave is a wave that is sustained in the perpendicular fluctuations of the electric and magnetic fields, the equation of the wave is
E (y) = Eo cos (kx –wt)
B (z) = Bo cos (kx-wt)
Let's look for the terms to build these equations. The speed of the wave is given by
c = λ f
The frequency and period are related
f = 1 / T
Let's start by applying this equation our case
f = c /λ
f = 3 10⁸/225 10⁻⁹
f = 1.33 10¹⁵ Hz
The angular velocity and the wave number are
w = 2π f
k = 2π /λ
w = 2π 1.33 10¹⁵ = 8.38 10¹⁵ rad / s
k = 2π / 225 10⁻⁹ = 2.79 10⁷ m⁻¹
It indicates that the period increases by a factor of 1.5, let's look for the new frequency
T = 1.5 T₀
f = 1 / T
f = 1 / 1.5T₀
f = 1 / 1.5 f₀
f = 1 / 1.5 1.33 10¹⁵ = 8.87 10¹⁴ Hz
c = λ f
λ = c / f
λ = 3 10⁸ / 8.87 10¹⁴ = 4,229 10⁻⁸ m
Let's find the new w and k
w = 2π f
w = 2π 8.87 10¹⁴ = 5.571 10¹⁵ rad/s
k = 2π / λ
k = 2π / 4,229 10⁻⁸ = 1,486 10⁸ cm⁻¹
We use the relationship that the fields are in phase
c = E₀ / B₀
B₀ = E₀ / c
B₀ = 7.4 10⁻³ / 3 10 ⁸ = 2.467 10⁻¹¹ T
With these values we can build the equation of the magnetic field
B (z) = 2,467 10⁻¹¹ cos (1,486 10⁸ x - 5.571 10¹⁵ t )