2nd Year Electromagnetism 2012: Practice Problems on ... 2nd Year Electromagnetism 2012: Homework...

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  • 2nd Year Electromagnetism 2012: Practice Problems on Vector Calculus.

    Lecturer: Steve Cowley

    These problems are optional and should be entirely revision – hopefully they will help you remember vector calculus from last year.

    (i) Let ψ = ψ0e −r2 where r2 = x2 + y2 + z2 is the radius in spherical polar coordinates and ψ0 is a constant.

    Show that

    ∇ψ = −2rψ0e−r 2

    (1)

    where r = xi+ yj+ zk

    (ii) Let ψ = ψ0 sin(kx)e −ky where k and ψ0 are constants. Show that

    ∇2ψ = 0 (2)

    (iii) Let ψ = ψ0 1 r sin(kr − ωt) where: r is the radius in spherical polar coordinates, ψ0 is a constant, the

    constants k and ω are related by ω = kc and, c is the velosity of light . Show that ψ satisfies the wave equation:

    c2∇2ψ = ∂ 2ψ

    ∂t2 (3)

    (iv) Find the curl of the vector field A = yi − xj. Draw the ”field lines” of A. Show that we can write A = ∇ψ × k and find ψ.

    (v) Consider a vector field A(x, y, z) = ∇a ×∇b where a = a(x, y, z) and b = b(x, y, z) are arbitrary scalar fields. Show that ∇ ·A = 0. Also find C such that ∇×C = A. Is C unique?

    (vi) Let P = f(r · d)(r× d) where r = xi+ yj+ zk and f is an arbitrary function. Show that ∇ ·P = 0.

    (vii) Let ψ = ψ0e ik·r where ψ0 is a constant and k is a constant (wave) vector. Show that ∇ψ = ikψ.

    (viii) Let A = A0e ik·r where A0 is a constant vector. Show that ∇ ·A = ik ·A and ∇×A = ik×A.

  • 2nd Year Electromagnetism 2012: Homework 1.

    Lecturer: Steve Cowley

    Question 1. Dipoles. This should be entirely revision – but hopefully it will get you warmed up. Consider a (model) polar diatomic molecule with charge +q at +∆r and −q at −∆r.

    (i) Give an exact expression for the electric field. For this molecule we define the dipole moment as p = 2q∆r. (ii) Give a simple expression for the electric field when ∆r ≪ r – write your answer in terms of p. (iii) Describe qualitatively what would happen to two isolated dipoles placed near each other – how would they move?

    Question 2. Spherical Charge Distributions.

    Let the charge density ρ be given by: ρ = ρ0 for r < a and, ρ = 0 for r > a where ρ0 = constant, a= constant and r =

    √ x2 + y2 + z2. Use Gauss’s law to find the electric field E for r < a and r > a.

    Question 3. Cylindrical Charge Distributions.

    Let the charge density ρ be given by: ρ = ρ0 r a for r < a and, ρ = 0 for r > a where ρ0 = constant, a=

    constant and r = √ x2 + y2. Use Gauss’s law to find the electric field E for r < a and r > a.

    Question 4. Electric Field of Arbitrary Stationary Charge.

    In this question we derive the formula for the electric field from a stationary distribution of charge. (i) Consider a spherical distribution of charge; i.e. ρ = ρ(r) with r =

    √ x2 + y2 + z2. Show that in this case,

    ∇ ·E = 1 r2

    ∂r (r2Er) =

    1

    ϵ0 ρ(r). (1)

    where Er is the radial component of E. What about Eθ and Eϕ? (ii) Integrate to find a formal expression for Er. (iii) Suppose ρ = 0 for r > a. Express Er for r > a in terms of the total charge. (iv) Let

    ρ = ρ0 = 2q

    4πrr20 e (− r2

    r2 0

    ) . (2)

  • Find an explicit expression for Er. Find Er in the limit r0 → 0.

    In the limit r0 → 0, ρ0 is essentially zero everywhere except close to the origin where it tends to infinity. Thus in this limit ∫

    V

    ρ0A(r)dV → A(0) ∫ V

    ρ0dV = A(0)q. (3)

    if the volume V contains the origin. We can define the three dimensional Dirac delta function as:

    δ(r) = lim r0→0

    ( 2

    4πrr20 e (− r2

    r2 0

    )

    ) . (4)

    We can therefore write the charge density of a point charge as ρ = qδ(r).

    (v) Argue that for a point charge q at r = r′ we may write ρ = qδ(r− r′).HARDER (vi) Find E = G(r, r′) for ρ = ϵ0δ(r− r′) i.e. find the solution to:

    ∇ ·G = δ(r− r′). (5)

    We often refer to functions like this as Green’s functions. HARD (vii) Hence show that,

    E(r) =

    ∫ ρ(r′)

    (r− r′) 4πϵ0|r− r′|3

    dV ′ (6)

    where the integration is over all r′ and |r− r′| is the distance between r and r′.

    Question 5. Magnetic Field from a Steady Current.

    (i) Calculate the magnetic field inside and outside a long straight conductor with circular cross section of radius R that carries a uniform current density with total current I.

    (ii) Now suppose that inside the conductor, there is a cylindrical hole of radius a whose axis is parallel to the axis of the conductor and a distance b from it (a + b < R). Assume again that the current density is constant and that the total current is I. Show that the magnetic field inside the hole is uniform and equal

    to µ0bI

    2π(R2 − a2) . (7)

    Draw a picture showing the orientation of the fields and the hole.

  • Question 6. Coulomb Explosions.

    In this question you are expected to make estimates – the more accurate the better. In the basement of Blackett high powered lasers were used to blast the electrons off clusters of a few thousand atoms. Suppose all the electrons are suddenly removed from a cluster of one thousand carbon atoms. The carbon nucleii then explode outwards – why and how fast? Hint consider the energy. How does the energy of the explosion scale with the size of the cluster?

  • 2nd Year Electromagnetism 2012: Homework 2.

    Lecturer: Steve Cowley

    Question 1. Solenoids. In this question the integral form of ampere’s law

    (i) Show that the field inside a long cylindrical solenoid is B = µ0nI where n is the number of turns per unit length and I is the current in each turn (see above). A thin solenoid of length 10cm with 600 turns carries a current of 20 amperes, calculate the field inside the solenoid.

    (ii) Now suppose that the solenoid is wrapped into torus – a doughnut shape – this is sometimes called a toroidal solenoid. Calculate the field inside the torus. If you want help see

    http://www.youtube.com/watch?v=jdsUQs9w0uw

    (iii) The fusion experiments JET and ITER (http://www.iter.org/) are based on huge toroidal solenoids. Schematically shown above. The field created by the 18 superconducting ”toroidal field coils” (like the blue ones above) in ITER is 5.2 tesla at a radius of about 6 meters. What is the current in each toroidal field coil?

    Question 2. Charge Conservation.

    (i) Deduce charge conservation from Maxwell’s equations. (Former exam question).

    (ii) Suppose magnetic monopoles (magnetic charges) exist and that magnetic charge is conserved. Let ρB be the density of magnetic charge and JB the current density of magnetic charge. We take units so that,

    ∇ ·B = µ0ρB . (1)

  • Write down the equation for the conservation of magnetic charge.

    (iii) Modify Faradays law to make Maxwell’s equations conserve magnetic charge.

    Question 3. Induction and Transformers.

    Consider a 10cm long solenoid (solenoid A) of radii approximately 1cm with 400 turns.

    (i) Let the current in the A, IA, change with time slowly (IA = I0(t)) so that you may ignore the displacement current. What is the instantaneous magnetic flux in the solenoid?

    (ii) A voltmeter is connected across the solenoid. The meter will measure ∫ E · dl integrated along the

    wire. Derive an expression for the voltage measured (ignoring the resistance). Calculate the voltage when I0 = sin [100πt]

    Consider now two 10cm long solenoids of radii approximately 1cm: solenoid A with 400 turns and solenoid B with 800 turns. B is fitted snugly inside A see diagram below.

    B  

    B  

    VA  

    VB  

    Double  solenoid  –  solenoid  B  inside  solenoid  A  

    IB  

    IA  

    (iii) Let current IA flow in A and current IB flow in B. What is the field inside both?

    Now we disconnect B and connect it to the volt meter. We put a slowly changing current IA = I0(t) through the coil A.

    (iv) Calculate the induced voltage across B.

    (iii) In the general case where both IA and IB are changing show that the voltages measured across A, VA, and across B, VB , satisfy VB/VA = 2.

    Question 4. Pinch Effect.

    MAGPIE in the basement of Blackett squeezes plasmas with magnetic fields and makes very dense and hot

  • plasmas. In this question you will estimate the forces involved.

    (i) Show that if we assume the current in the (neutral) plasma is carried by many electrons and that the ions are stationary then

    J = −neev (2)

    where ne is the electron number density (the number of electrons per unit volume) and v is the average velocity of the electrons.

    (ii) Recall that the force on a single electron is −e(E+ v ×B). Hence show that the force per unit volume on the electrons and ions in the plasma (and hence on the plasma) is:

    J×B (3)

    if we assume that the plasma is neutral so that the electron and ion charge densities acancel i.e. ene = qni where q is the ion charge.

    (iii) Magpie makes a cylindrical shell of current flowing in plasma made from vaporized wires. The current is taken to be zero except in a shell ofradius 0.01 metres and thickness 0.001 metres where:

    J = J0z J0 = constant. (4)

    Calculate the magnetic field (ignoring displacement current).

    (iv) Calculate