suggested problems for ovning - jfuchs.hotell.kau.se fileelektromagnetiskfalt¨...

ELEKTROMAGNETISK FALTTEORI, VT 2015, FYGB03 /FYGB09

Suggested problems for ovning 3 (5/2) :

1. Two charged point masses, of equal mass m and equal charge q , are suspended bythreads of equal length a (and negligible mass) at a common point.

Compute the angle α between the two threads when the system is in equilibrium.

2. Problem P.3–8 of [Cheng] (uniform line charge forming a semicircle, page 146).

3. Determine the electricfield ~E of an (infinitely thin) circular ring of radius b anduniform line charge ρl , for all points on the straight line that is normal to the centerof the circle.

At which points on the axis is the magnitude of ~E maximal ?

Consider the limit that the radius b is taken to zero and the line charge ρl isincreased in such a way that the total charge Q remains constant.

4. Problem P.3–11 of [Cheng](spherical charge distribution surrounded by conducting shell, page 146).

5. Problem P.3–15 of [Cheng](three charges q, −2q, q arranged symmetrically, page 146).

6. What is Dirac’s delta ‘function’ ?

(It’s a good idea to search the literature, or the web, for information about thedelta function.)

List its most important properties.

Express the three-dimensional delta function in terms of one-dimensional delta func-tions. Give the result in Cartesian, cylindrical and spherical coordinates.

FYGB03 / FYGB09 – VT 2015 1 2015-03-09



7. The (time-average) electric potential of a hydrogen atom is approximately given by

V = |Qe|e−αr

4πǫ◦r

(

1 +αr

2

)

.

Find a charge distribution (having both a continuous and a discrete part) thatresults in such a potential.

Interpret the result. What is the meaning of the parameter α ?

8. Compute the electrostatic energy that is stored in a uniformly charged ball of radiusR .

Also (try to) do the analogous computation for a cube of edge length a .

9. Compute the force due to a point charge Q at position ~a on an electric dipole thatconsists of point charges +q located at ~b and −q located at −~b, for ~a×~b=0and |~b|≪ |~a| .

Express the result through the dipole moment of the electric dipole.

10. Problem P.3–23 of [Cheng] (electric field in a spherical cavity, page 148).

11. Problem P.3–25 of [Cheng] (interface between two lossless dielectrics, page 148).

12. Problem P.3–27 of [Cheng] (boundary conditions for the electricpotential at aninterface between dielectrics, page 148).

13. Problem P.3–46 of [Cheng] (partially filled parallel-plate capacitor, page 150/151).

FYGB03 / FYGB09 – VT 2015 2 2015-03-09



14. The outer conductor of an infinitely long coaxial cable is grounded, while the core(inner conductor) of the cable is held at a constant potential Vi .The radius of the core is bi . The inner radius of the outer conductor is bo .

• Determine the electric potential V (~r) at every point between the coreand the outer conductor.

• Determine the surface charge density ρs on the surface of the core.

• Compute the capacitance per unit length of the cable.

Hint: Use cylindrical coordinates.

15. Consider the continuous charge distribution ρ(~r) that in spherical coordinates(r, ϑ, ϕ) has the form

ρ(~r) =

ρ0 r

bfor b≤ r≤ 2b ,

0 for r < b or r > 2b ,

with constant ρ0 and b .

Determine, for every point ~r , the electrostatic potential V (~r) and the electric~E -field ~E(~r) that result from this charge distribution.

16. A long wire with conductivity σ and with circular cross section of radius b – calledthe core – is surrounded by a circular layer of a material with conductivity 1

3σ –

the coating .

A total steady current I is running in the coated wire.

• How thick must the coating be in order that the resistance of the coated wire is50% of that of the core ?

• Determine the electric ~E -field both in the core and in the coating.

FYGB03 / FYGB09 – VT 2015 3 2015-03-09



17. Determine the magnetic ~B -field at every point in the interior of an infinitely longcoaxial transmission line for which both the inner and the outer conductor havefinite thickness.

18. Compute the magnetic ~B -field at an arbitrary point on the axis of a closely woundcircular solenoid of radius a and finite length ℓ , carrying a steady current I .

What is the ratio between the values of | ~B| at the center and at the ends of thecoil ?

Check that in the limit ℓ→∞ the known result for an infinitely long solenoid isreproduced.

19. Problem P.6–12 of [Cheng]

(Helmholtz coils).

20. Problem P.6–15 of [Cheng]

(obtaining a homogeneous ~B-field by digging a hole).

FYGB03 / FYGB09 – VT 2015 4 2015-03-09



21. Example 6–15 of [Cheng] :

Determine the inductance L′ per unit length of a very long solenoid (with air core,and of uniform winding).

22. Example 6–18 of [Cheng] :

Compute the mutual inductance between two coils that are wound (with N1 andN2 turns, respectively) on a straight cylindrical core of permeability µ .

23. Show that the magnetic field

~B(~r) =

{

B0 ~ez for r < b ,

B0 (br)3(cosϑ~er +

12sinϑ~eϑ) for r > b (spherical coordinates)

with constant B0 satisfies the fundamental equations of magnetostatics and com-pute the electric current density that generates this field.

a) In the region 0 < r < b .

b) In the region r > b .

c) For r = b .

24. A simple-minded model for the origin of Earth’s magnetic field is to imagine thatit results via Earth’s rotation from some homogeneous surface charge density.Assume that this provides indeed a valid picture of Earth’s magnetic field.

Hint: Before solving this problem, first have a look at the solution of problem 23 .

For parts c) – e) you may use the information that the corresponding magnetization is

uniform, even if you are not able show this.

a) What sign must the surface charge density have ?

b) Show that the magnetization inside the Earth that would correspond to such

a current density is uniform.

c) Derive an expression for the magnetic dipole moment of the Earth.

d) Calculate the magnitude of the surface charge density.

e) What does the resulting electric field look like ?

From the result, draw conclusions on the possible validity of the model.

FYGB03 / FYGB09 – VT 2015 5 2015-03-09



25. Consider a Wheatstone bridge – that is, an electric circuit of the following form:

Four resistances R1, R2, R4, R3 connected in series, with points A (between R1 andR2 ) and C (between R4 and R3 ) connected to a battery and points B (betweenR2 and R4 ) and D (between R3 and R1 ) connected to a Voltmeter (which hasnegligible resistance).

a) Show that when the voltage V at the Voltmeter is zero, then

R1/R3 = R2/R4 .

b) Suppose that R1 increases by a factor 1+x with x≪ 1 . Set

R3 = aR1 , R2 = bR1 , R4 = abR1 .

Find the voltage V measured by the Voltmeter as a function of x, a, b.

c) What can this circuit possibly be used for ?

When is its sensitivity maximal ?

26. A (test) particle of charge q is moving in a static homogeneous magnetic field ~B .

a) Show that the motion describes a helix whose axis is given by the direction of

the field ~B .

b) Determine the angular velocity of the circular part of the motion.

c) How does the radius of the circle depend on the magnitude of ~B ?

d) Discuss what the motion looks like when in addition there is also a static homo-

geneous electricfield ~E .

27. Problem P.7–6 of [Cheng](reduction of eddy-current power loss in a transformer, page 349).

28. Problem P.7–14 of [Cheng](wave equations in an inhomogeneous isotropic medium, page 351).

FYGB03 / FYGB09 – VT 2015 6 2015-03-09


Suggested problems for ovning 8.5 (13/3) :

29. a) Use the two Maxwell equations that involve the divergence of ~B and the curl of~E to express the fields in terms of a vector potential ~A and a scalar potential V .

b) Assume that the magnetic vector potential is initially given explicitly as some

vector field ~A◦(~r). Replace it according to

~A◦ 7→ ~A(1) := ~A◦ + ~∇U ,

with U an arbitrary scalar function, by the field ~A(1) .

What changes in the magnetic ~B -field and the electric ~E -field result from thischange of ~A ?

c) Modify analogously also the scalar potential V ,

V◦ 7→ V(1) .

What must this modification look like in order that it exactly compensates thechange of ~E that comes from the change of ~A ?

d) Show that after substituting the result of part a) into the other two Maxwell

equations one obtains uncoupled wave equations for the potentials ~A and V pre-cisely if the Lorentz condition

~∇· ~A+ µǫ ∂V∂t

= 0

is fulfilled.

Explain why one is allowed to impose this gauge condition.

e) What condition must the function U that appears in part b) – and in part c) ,

too – satisfy in order that both the original potentials ~A◦ and V◦ and the newpotentials ~A(1) and V(1) obey uncoupled wave equations ?

30. Compute the self-inductance of a circular loop made out of an infinitely thin wire.

FYGB03 / FYGB09 – VT 2015 7 2015-03-09



31. Problem P.6–34 of [Cheng] (current-carrying wire parallel to an infinite planarinterface, page 303).

32. Consider an initially uncharged conducting ball of radius b that is placed in aninitially uniform electric field ~E◦ .

a) Compute the resulting electrostatic potential V (~r) and electric field ~E(~r) ev-erywhere in space.

Hint:

Describe the uniform field ~E◦ as the field of an electric dipole consisting of point charges

q and −q separated by a distance d , in the limit that both q and d tend to infinity in

such a way that the ratio q/d2 remains constant.

Then obtain the total electric field ~E(~r) as the superposition of the field of this dipole

and of the field of the corresponding image charges.

b) Determine the electric surface charge density that is induced on the surface ofthe ball.

c) What is the value of the total electric surface charge on the ball ?

33. Consider the half-space given by x≥ 0 , −∞<y, z <∞ .

On the boundary the following values of the potential V are prescribed:{

V (~r)→ 0 for |~r| → ∞ ,

V (0, y, z) = (a2+ y2+ z2)−1/2/4πǫ◦ with a> 0 constant .

Determine the potential everywhere in the half-space

a) when there are no charges in the half-space;

b) when the half-space contains a point charge at ~r= a~ex .

Hint: Solve part b) before part a) .

34. Estimate the following quantities for a ‘typical’ thunderstorm:

(1) The capacity between Earth and the clouds.

(2) The total charge, toal current, and total energy.

FYGB03 / FYGB09 – VT 2015 8 2015-03-09

suggested problems for ovning - jfuchs.hotell.kau.se fileelektromagnetiskfalt¨...

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