E&M 2

Maxwell's equations (SI units)

·E = ρ/ε0×E = -∂B/∂t,
·B = 0,  ×B = μ0j + (1/c2)∂E/∂t,
or, in macroscopic form,
·D = ρf×E = -∂B/∂t,
·B = 0,  ×H = jf + ∂D/∂t.

In lih materials with D = εE and B =  μH and  ε, μ = constant we have
·E = ρf/ε,  ×E = -∂B/∂t,
·B = 0,  ×B = μjf + εμ∂E/∂t.

The method of images

Consider a charge q at  z = d on the z - axis.
Assume the z = 0 plane is a grounded conducting plane.  Then placing an image charge 
q' = -q at  d' = -d on the z-axis makes the z = 0 plane an equipotential with Φ = 0. 
The electrostatic field and potential above the plane are the same as the field and potential due to the charge and its image charge.

AC circuits

Assume V(t), I(t), ε(t) are all proportional to exp(iωt).
Assume idealized circuit elements.  Define the impedance Z = V/I.  Then
Z(capacitance) = ZC = 1/(iωC),
Z(inductance) = iωL,
Z(resistance) = ZR = R.
Any impedance may be written as Z = R + iX.

Kirchhoff's rules for filamentary circuits:  

For each loop ∑nVn = 0  for each node  ∑nIn =0.
Any two-terminal network of passive elements is equivalent to an effective impedance Zeff
Thevenin equivalent circuits:  Any two terminal network can be replaced by a generator emfeff in series with an impedance Zeff.


Problem 1:

A positively charged particle is moving in the xy-plane on a region where there is a non-zero, uniform magnetic field B in the +z direction and a non-zero, uniform electric field E in the +y direction.  Which of the following is a possible trajectory for the particle?

image

Problem 2:

A region of space contains a uniform electric field in the +z-direction and a uniform magnetic field in the -z direction.  Which of the following statements is true?

(A)  A particle could have a positive charge and an initial velocity, so that it would pass unaccelerated through this region.
(B)  The Poynting vector of this distribution of fields is zero.
(C)  This arrangement of fields can produce the Hall effect.
(D)  The vector potential of the magnetic field is also parallel to the z-axis.
(E)  Since the divergence and curl of both fields are zero, the fields must be anti-parallel everywhere in space.

Problem 3:

If an electric field is given in a certain region by Ex = 0, Ey = 0, Ez = kz, where k is nonzero, which of the following is true?

(A)  There is a time-varying magnetic field.
(B)  There is charge density in the region.
(C)  The electric field cannot be constant in time.
(D)  This electric field is impossible under any circumstances.
(E)  None of the above.

Problem 4:

A rectangular loop of wire with resistance R has the dimensions shown. 

image

A long wire with current I is located in the plane of the loop a distance r from one side of the loop.  If the loop is pulled radially away from the wire at constant speed, the current in the loop is

(A)  [μ0Ivab/(2πr)] (1/r)    (B)  [μ0Ivab/(2πr)] (1/r2)   (C)  [μ0Ivab/(2πr)] (1/(r+a))   

(D)  [μ0Ivab/(2πr)] (1/(r(r+a)))    (E)  [μ0Ivab/(2πr)] (1/(r2(r+a)))

Problem 5:

A capacitor is charged with two batteries as shown.  If initially there is no charge on C, which of the following expressions gives the time dependence of Q on C?

image

(A)  Q0[1 - exp(-t((R1 + R2)C)]    (B)  Q0[1 - exp(-t(R1 + R2)/(R1R2C))]    (C) Q0exp(-t(R1 + R2)/(R1R2C))   

(D)  Q0exp(-t(R1 + R2)C)   (E)  Q0exp(-t/((R1R2)1/2C))

Problem 6:

A point charge q is located a distance d above a grounded conducting plane. 

image

The magnitude of the electric field at point P just above the surface is

(A)  [1/(4πε0)] q/R2    (B)  [1/(4πε0)] 2q/R2    (C)  [1/(4πε0)] 2qd/R3   

(D)  [1/(4πε0)] 2qx/R3    (E)  [1/(4πε0)] 2q/(xd2)

Problem 7:

When the bridge is balanced by adjusting Rs, which of the following statements is false?

image

(A)  The current through the galvanometer is zero.
(B)  i2 = ix.
(C)  isRS = ixRx.
(D)  The potential at A is the same as that at B.
(E)  The potential at C is the same as that at D.

Problem 8:

A certain electric field is given in spherical coordinates by E(r) = A r-(3/2) (r/r), where (r/r) is a unit vector radially outward and A is a constant.
The charge density ρ is proportional to

(A)  r1/2   (B)  r-1/2   (C)  r-3/2   (D)  r-5/2    (E)  r-5/2

Problem 9:

Which of the following equations is a consequence of ×H = jf + ∂D/∂t.

(A)  ·(jf + ∂D/∂t) = 0    (B)  ×(jf + ∂D/∂t) = 0   

(C)  (jf·∂D/∂t) = 0    (D)  jf + ∂D/∂t = 0    (E)  jf·∂D/∂t = 0

Problem 10:

Which of the following pairs of macroscopic electromagnetic fields is continuous across a boundary between two media of different dielectric constants and different magnetic permeabilities?

(A)  B and BN    (B)  B and EN    (C)  D and BN    (D)  E and HN    (E)  H and DN