When a polarized beam travels through a polarizing sheet, its new intensity is

I = I_0 * cos^2(Δθ)

where Δθ is the difference in angle between the beam's polarization and the sheet's polarization. As you can see, the intensity will always be reduced. Obviously we want the total angle change to be 90 degrees, but we can achieve this in smaller increments of Δθ. After applying n sheets, the reduction will be

I / I_0 = cos(Δθ)^2n

and nΔθ = 90 degrees. Solve for the number of sheets n that gives you an intensity of 60% the original.

This problem is easiest if you solve in terms of variables and plug in the 2.0 A at the very end. Start with the expression for the displacement current. Can you write the electric flux as the field E times an area? You also know the expression for a parallel plate capacitor. What is E inside? Remember that the discharging current I = dQ/dt, where Q is the charge on the capacitor.

This should be enough to solve the problem, but you have to be careful when you write down areas: there are two different areas involved -- the entire plate area and the center circular area.

You can call the current I and solve the problem as if you knew it. You are trying to find the radius at which the magnetic field is half its max value, so you don't need to know exactly what the values are.

A discharging parallel-plate capacitor creates a "displacement current" within the gap, which produces a magnetic field. You know the displacement current to be:

I_d = ε_0 dΦ/dt

where Φ is the electric flux. Use the equation for the electric field of a parallel plate capacitor and remember that current I = dQ/dt. From here you can find the flux and dΦ/dt to find the displacement current.

Now you can treat the displacement current like any other and use Ampere's law to find the magnetic field. (Draw an Amperian loop at some arbitrary radius)

I'm not sure what part you are stuck on, so I will try to give you tips throughout.

Use the power and distance to find the flux/time/area at 20 m away

Because the sphere is small compared to the distance, approximate it to be a circle facing the source

Find the flux on the circle using the flux/area you found

Now is where you have to think. The units of what you have are Energy/time. force is dp/dt.

You have Energy transferred to the sphere per time and you want momentum transfer per time.

Knowing this is light, can you get the momentum from known energy?

Hope this helps!

Student 5051 is correct. integrating and solving for the magnetic field at R' should give:

B= (1/3)*mu*epsilon*(3R'/2 - R'^2/R)

Good luck!

Keep in mind that while current loops created a dipole moment, they are not point-dipoles or "perfect dipoles." At a large distance away, the fields look just like perfect dipole, but up close they do not.

Hope this helps!

The potential energy difference corresponds the total spin up and spin down configuration inside the material.

Recall that the equation for flux is ∮_A B⃗⋅ n̂ dA. Further recall that the B⃗ for a solenoid is constant within the solenoid and zero outside. Look at evaluating this integral across the area of the larger solenoid cross-sectional face. Where is B⃗ zero? Where is it not zero? How does this affect the integral?

Hope this helps!

In general you never want to assume you know the direction of the force of a wall. It is best to write it in terms of its components Fx and Fy. You can treat these components as if they were two separate forces pointing in the x and y directions respectively, and at the end you can recombine them into the total force of the wall to find the direction and magnitude.

Draw a picture, then draw all the forces acting on the beam. Thinking it through we have:

- Force of gravity on the beam, pointing down

- Fx and Fy from the wall

- Suspended weight, pointing down

- Tension from cable, pointing along cable

You know the net force and net torque must be zero, so set the x and y components of the net force to zero and set the sum of the torques to zero.