PHYSICS 401 – PROBLEM SET # 1

Due January 16, 2008, (due in class)

This first assignment is a set of review problems, just to be sure you are brushed up onthe material in PHYS 301/354.

Please review Chapters 1-7 in textbook:David Griffiths, Introduction to Electrodynamics, 3rd Edition,

1. a) Starting with Maxwell’s equations in differential form, derive Maxwell’s

equations in integral form.

b) Starting with Maxwell’s equation

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r ∇ ×

r B = µ0

r J + µ0ε0

∂r E ∂t

, derive the

continuity equation

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r ∇ ⋅

r J + ∂ρ

∂t= 0 , which is the mathematical expression

of conservation of charge.c) Starting with Maxwell’s equations in free space, (

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r J = 0, ρ = 0), show

that

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r E and

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r B each satisfy a wave equation. What is the velocity of

propagation of the resulting wave in each case?

2. Two spherical cavities, of radii a and b, are hollowed out from the interior of asolid neutral conducting sphere of radius R, as shown in the figure.

a) What is the electric field in the solid (shaded)conducting material?

b) Find the surface charges

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σ a, σ b , and

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σR , at therespective surfaces.

c) What is the electric field outside the conductor atdistance r>R?

d) What is the electric field inside cavity a?… inside cavity b?

e) What is the force on qa ? … and on qb?f) If a third charge, qc were brought near the

conductor, would the following quantities staythe same or change?

i.

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σ a ii.

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σ b iii.

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σR iv. the electric field inside cavities a and b v. the electric field outside the conductor

3. A capacitor is constructed of two very long concentric cylindrical conductors,with the space between them exactly half filled with linear dielectric, withdimensions as shown in the diagram, and with dielectric constant κ.

a) Show that if free charge was placed on the coaxial conductors, theresulting electric field would be radial and is the same in the dielectric andin the vacuum halves of the capacitor.

b) Deduce the capacitance per unit length of this coaxial capacitor.c) If the conductors carry free charges per unit length ±λ, find the

polarization

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r P in the dielectric at any point a distance r from the central

axis, in terms of

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ε0 , κ, λ and r.

4. Two very long parallel wires carry equal currents I in opposite directions, asillustrated in the figure. Take the

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ˆ z axis direction to be out of the page, in thedirection of the current in wire 1. The field point P is located a distance r1 fromwire 1 and distance r2 from wire 2, as shown.

a) Consider each wire separately and indicatethe direction of the vector potential

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r A in

each case.b) Show that the vector potential

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r A at the

point P is given by:

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r A = µ0I

2πln r2

r1

ˆ z

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Head on view

c) Show that the result in part b) is consistentwith that obtained using Ampère’s Law.

3-D view

5. Consider an infinitely long metal pipe, of radius R, which is placed at right anglesto an otherwise uniform electric field

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r E 0 = E0 ˆ x .

a) What is the “uniqueness theorem” and why would you want to use it tosolve for the electric potential V?

b) What are the boundary conditions on the electric potential, V?c) Solve Laplace’s equation for the potential V outside the long metal pipe.

Hint: Note that this situation has cylindrical symmetry (not spherical!),with no z dependence, and hence simplifies to a 2-D plane polar problem.You should obtain:

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V (r,θ) = E0 rR2

r2−1

cosθ

6. A square loop of wire of side s is near a very straight long wire which is carryingcurrent I, as shown in the figure.

a) Find the magnetic field due to the current carrying wire.b) Find the flux of

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r B through the loop.

c) If the loop is pulled directly away from the wire (upwards in the diagram)at speed v, what is the emf generated?

d) Does the induced current flow clockwise or counter-clockwise in the loop?e) What is the induced current in the loop if the loop is pulled directly to the

right, instead of upwards?