Problem 1 (10 marks ) Consider a wire loop, with resistance R and radius ra, in a magnetic field as shown in Figure 1. At some time t1 the flux of the magnetic field through the loop is given by �1....

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Problem 1 (10 marks ) Consider a wire loop, with resistance R and radius ra, in a magnetic field as shown in Figure 1. At some time t1 the flux of the magnetic field through the loop is given by �1. Between time t1 and a later time t2 the flux changes to �2. Determine the total charge that flows through the loop over the time interval between t1 and t2. Problem 2 (20 marks) Figure 2 shows a conducting spherical shell with inner radius Rb and outer radius Rc that encloses a concentric solid conducting sphere of radius Ra. The solid sphere has a charge of Q. Determine the capacitance of the spheres. Problem 3 (20 marks) Consider the two wire loops shown in Figure 3. Loop 1 has length L and width h and carries a current I. Loop 2 has length ` and width a and is located a distance b from Loop 1, in the same plane as Loop 1. Determine the mutual inductance of the two loops. You may assume that the sides of Loop 1 of length L are very long and that L � `. 1 Problem 4 (25 marks) Consider two thin cylindrical conducting shells as shown in Figure 4. The inner shell has a radius Ra, length L, and charge Q1 uniformly distributed over its surface. The outer shell has a radius Rb, length L, and charge Q2 uniformly distributed over its sur- face. The two shells are rotating in opposite direc- tions with the same angular velocity !. As seen from above, the outer shell is rotating in a clockwise direc- tion. (a) Determine the surface current densities K1 and K2. (10 marks) (b) Find the magnetic field B in the three regions: (i) s < ra,="" (ii)="" ra="">< s="">< rb,="" and="" (iii)="" s=""> Rb. You can assume that the cylinders are very long and neglect edge e↵ects. (15 marks) Problem 5 (25 marks) Figure 5 shows an infinitely long cylindrical wire with radius Rw in which the current density in- creases over a time interval ⌧ as follows: J(s, t) = 8 >>><>>>: 0 for t  0 J0 s t/⌧ ẑ for 0  t  ⌧ J0 s ẑ for t � ⌧ where J0 is a constant and is positive. Here s is the distance from the center of the cylindrical wire. At a distance b (with b > Rw) from the center of the cylindrical wire is a square wire loop with sides of length a and resistance R. Both the cylindrical wire and the square loop are fixed in space. (a) During the interval 0  t  ⌧ , find the magnetic field B of the long cylindrical wire in the two regions: (i) s < rw="" and="" (ii)="" s=""> Rw. You may neglect inductance e↵ects in answering this part of the problem. (10 marks) (b) Determine the magnitude and direction of the current induced in the square wire loop during the interval 0  t  ⌧ . (10 marks) (c) Determine the magnitude and direction of the net force on the square wire loop during the interval 0  t  ⌧ . (5 marks) 2 Possibly Useful Constants and Equations Cartesian Coordinates dl = dx x̂+ dy ŷ + dz ẑ; d⌧ = dx dy dz Gradient: rt = @t @x x̂+ @t @y ŷ + @t @z ẑ Divergence: r · v = @vx @x + @vy @y + @vz @z Curl: r⇥ v = ✓ @vz @y � @vy @z ◆ x̂+ ✓ @vx @z � @vz @x ◆ ŷ + ✓ @vy @x � @vx @y ◆ ẑ Laplacian: r2t = @ 2t @x2 + @2t @y2 + @2t @z2 Spherical Coordinates dl = drr̂+ rd✓ ✓̂ + r sin ✓d��̂; d⌧ = r2 sin ✓ dr d✓ d�; da = r2 sin ✓ d✓ d� r̂ 8 ><>: x = r sin ✓ cos� x̂ = sin ✓ cos�r̂+ cos ✓ cos�✓̂ � sin��̂ y = r sin ✓ sin� ŷ = sin ✓ sin�r̂+ cos ✓ sin�✓̂ + cos��̂ z = r cos ✓ ẑ = cos ✓r̂� sin ✓✓̂ 8 >><>>: r = p x2 + y2 + z2 r̂ = sin ✓ cos� x̂+ sin ✓ sin� ŷ + cos ✓ ẑ ✓ = tan�1 ⇣p x2 + y2/z ⌘ ✓̂ = cos ✓ cos� x̂+ cos ✓ sin� ŷ � sin ✓ ẑ � = tan�1(y/x) �̂ = � sin� x̂+ cos� ŷ Gradient: rt = @t @r r̂+ 1 r @t @✓ ✓̂ + 1 r sin ✓ @t @� �̂ Divergence: r · v = 1 r2 @ @r (r2vr) + 1 r sin ✓ @ @✓ (sin ✓v✓) + 1 r sin ✓ @v� @� Curl: r⇥ v = 1 r sin ✓  @ @✓ (sin ✓v�)� @v✓ @� � r̂+ 1 r  1 sin ✓ @vr @� � @ @r (rv�) � ✓̂ + 1 r  @ @r (rv✓)� @vr @✓ � �̂ Laplacian: r2t = 1 r2 @ @r ✓ r2 @t @r ◆ + 1 r2 sin ✓ @ @✓ ✓ sin ✓ @t @✓ ◆ + 1 r2 sin2 ✓ ✓ @2t @�2 ◆ 3 Cylindrical Coordinates dl = ds ŝ+ sd� �̂+ dz ẑ; d⌧ = s ds d� dz 8 ><>: x = s cos� x̂ = cos� ŝ� sin� �̂ y = s sin� ŷ = sin� ŝ+ cos� �̂ z = z ẑ = ẑ 8 ><>: s = p x2 + y2 ŝ = cos� x̂+ sin� ŷ � = tan�1(y/x) �̂ = � sin� x̂+ cos� ŷ z = z ẑ = ẑ Gradient: rt = @t @s ŝ+ 1 s @t @� �̂+ @t @z ẑ Divergence: r · v = 1 s @ @s (svs) + 1 s @v� @� + @vz @z Curl: r⇥ v =  1 s @vz @� � @v� @z � ŝ+  @vs @z � @vz @s � �̂+ 1 s  @ @s (sv�)� @vs @� � ẑ Laplacian: r2t = 1 s @ @s ✓ s @t @s ◆ + 1 s2 @2t @�2 + @2t @z2 Vector Identities A · (B⇥C) = B · (C⇥A) = C · (A⇥B) A⇥ (B⇥C) = B(A ·C)�C(A ·B) r(fg) = f(rg) + g(rf) r(A ·B) = A⇥ (r⇥B) +B⇥ (r⇥A) + (A ·r)B+ (B ·r)A r · (fA) = f(r ·A) +A · (rf) r · (A⇥B) = B · (r⇥A)�A · (r⇥B) r⇥ (fA) = f(r⇥A)�A⇥ (rf) r⇥ (A⇥B) = (B ·r)A� (A ·r)B+A(r ·B)�B(r ·A) r · (r⇥A) = 0 4 r⇥ (rf) = 0 r⇥ (r⇥A) = r(r ·A)�r2A Fundamental Theorems R b a (rf) · dl = f(b)� f(a) R (r ·A) d⌧ = H A · da R (r⇥A) · da = H A · dl Electrodynamics r · E = 1 ✏0 ⇢ I E · da = 1 ✏0 Qenc r⇥ E = � @B @t r ·B = 0 r⇥B = µ0J+ µ0✏0 @E @t I B · dl = µ0Ienc B = r⇥A F = Q[E+ (v ⇥B)] E = �rV � @A @t r · J = �@⇢ @t V (r) = � Z r Ref E · dl E(r) = 1 4⇡✏0 Z ⇢(r0) r2 r̂d⌧ 0 E(r) = 1 4⇡✏0 Z �(r0) r2 r̂da 0 E(r) = 1 4⇡✏0 Z �(r0) r2 r̂dl 0 V (r) = 1 4⇡✏0 Z �(r0) r da 0 V (r) = 1 4⇡✏0 Z ⇢(r0) r d⌧ 0 V (r) = 1 4⇡✏0 Z �(r0) r dl 0 Eabove � Ebelow = � ✏0 n̂ W = 1 2 Z V ⇢V d⌧ = ✏0 2 Z all E2d⌧ C = Q V W = 1 2 CV 2 = 1 2 Q2 C W = 1 2 nX i=1 qiV (ri) I = dQ dt I = Z J · da J = ⇢v K = �v I = �v Fmag = I Z dl⇥B B(r) = µ0 4⇡ Z J(r0)⇥ r̂ r2 d⌧ 0 B(r) = µ0 4⇡ Z K(r0)⇥ r̂ r2 da 0 B(r) = µ0I 4⇡ Z dl0 ⇥ r̂ r2 A(r) = µ0I 4⇡ Z dl0 r A(r) = µ0 4⇡ Z K(r0) r da 0 A(r) = µ0 4⇡ Z J(r0) r d⌧ 0 J = �E V = IR � = Z B·da emf: E = I E·dl = �d� dt �1 = LI1 �2 = M21I1 5 Magnetic field of a wire segment: B = µ0I 4⇡s (sin ✓2 � sin ✓1)�̂ Magnetic field of a very long solenoid: Bin = µ0nIẑ Bout = 0 Fundamental Constants ✏0 = 8.85⇥ 10�12 C2/N m2 µ0 = 4⇡ ⇥ 10�7 N/A2 c = 3.00⇥ 108 m/s e = 1.60⇥ 10�19 C m = 9.11⇥ 10�31 kg 6