Physic Question

1 Problem 1

a) The uniformly charged circle of radius R , total charge q is rotating around the axis that is perpendicul to the plane where the circle is lying and is passing

through its center with angular frequency !. Find its magnetic moment. b) The sphere of radius R with uniformly charged surface (surface charge

density �) is rotating with angular velocity !. Find the magnetic dipole moment of the sphere. Hint: divide the sphere onto a lot of small circles.

c) The ball of radius R with uniformly charged volume ( total charge q) is rotating with angular velocity !. Find the magnetic dipole moment of the sphere. Hint: divide the ball onto spherical shells.

d) We know from the experiment that the charge of the electron is equal

by magnitude to the charge of the proton. However, every experiment has an

uncertainty, and for this data the uncertainty is 10 �21

(i.e., we know for sure

that |qe|�|qp|

|qe| < 10�21, but we cannot say purely from the experimental data that

it is strictly zero). If the charge of the electron was actually slightly di↵erent

from the charge of the proton within this uncertainty, could it generate the

magnetic field of the Earth?

Hint: assume that due to this slight di↵erence the Earth is uniformly charged.

Pick the element from the periodic table you think is most common on Earth (or

take the hydrogen for the simplicity, although of course it is not the hydrogen

that is most common). Estimate the number of electrons. Assume that it is

equal to the number of protons. Find the total charge of the Earth. Assuming

it to be uniformly distributed, find the magnetic dipole moment using the pre-

vious part of the problem. Find the magnetic field on the pole, generated by

such magnetic moment. Compare it with the actual field of the Earth.

2 Problem 2

Let us introduce vector A(x, y, z) such that

A(x, y, z) = µ0 4⇡

Z j(x0, y0, z0)

p (x � x0)2 + (y � y0)2 + (z � z0)2

dx0dy0dz0 (1)

where j is the current density. a) Using the law of Biot and Savart show that B = r ⇥ A. Hint: Find the

curl of f(x, y, z)a where a is a constant vector and f is a (scalar) function. b) Rewrite Maxwell equations using A instead of B. Do you get any simpli-

fications? (Hint: Simplify combinations of r ). c)For simple geometry if curl of vector is 0 it can be written as gradient

of a function. Using this knowledge, redefine E to turn one of the remaining equations into identity too. What is this function that you just introduced?

Hint: write equations as something is equal to zero and collect the terms.

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3 Problem 3

The surface of the parabolic mirror in 2D is given by the curve y = ax2. Find the focal distance of the mirror.

4 Problem 4

N parallel polarizing planes are put in sequence. The angle between the po- larizing axis of the first one and the last one is ⇡/2. The angles of nth plane relative to the first is �n

a) If the intensity of the light that is passes through the first plane is I1, what is the intensity of the light passing through the last plane IN?

b)** If the number of planes is fixed show using Lagrange multipliers that

the maximal intensityIN occurs if �n = ⇡n 2N

. Hint: Introduce variable ↵n = �n � �n�1, and maximize the expression you obtain in the previous part with respect to ↵n under the condition

P ↵n = ⇡/2. To maximize the function under

condition, add �( P

↵n � ⇡/2), whee � is Lagrange multiplier, to the function and find when the derivative with respect to all ↵n and � are zero.

c)* Assuming that the angles satisfies (b) what is the maximal possible

intensity IN? Hint: show by taking derivative that intensity IN increases if you increase N. To find the intensity in the limit N ! 1 consider what happens when you double the number of planes. Denoting I(N) the passing intensity for N planes and I(2N) for 2N planes, in the limit N ! 1 you should have I(2N) = I(N) = I(1). Solving this you find I(1)

5 Problem 5

If the light can go from point A to point B by many di↵erent ways they all should take the same time (it follows from Fermat principle under certain limitations).

This is called tautochronism. Deduce thin spherical lens formula from this

principle. Hint: consider the ray that goes from source to image along the main

optical axis of the lens, and the ray that touches the edge of the lens (but also

from the same source to the same image). Equate their optical lengths. Assume

the thickness of the lens to be much smaller than its diameter, and diameter

to be much smaller than distance to the object and image. Both surface of the

lens assume to be spherical.

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