Phys 301 proofs

Question 1

Prove that

1/2µ §(B²)dt = ½ §(A · J)dt

Answer 1

15

Question 2

Biot-Savart Law (from Maxwell’s equations for Magnetostatics)

Answer 2

10

Question 3

Coulomb’s Law (from Maxwell’s Equations for Electrostatics)

Answer 3

5

Question 4

Continuity equation (from Maxwell’s equations in differential form)

Answer 4

2

Question 4

Prove that:

Ø₂₁ = M₂₁I₁

€₂ = -M₂₁ (dI₁/dt)

Answer 4

18

Question 5

Conservation of electric charge (from the CONTINUITY equation)

Answer 5

3

Continuity equation: ►·J = - dp/dt

Question 8

Maxwell’s equations in integral form (from Maxwell’s equations in differential form)

Answer 8

1

Question 8

For two loops, prove that:

M₁₂ = M₂₁ = µ/4𠧧 (dl₁·dl₂)/|r₁-r₂|

Answer 8

17

Question 9

Work required to assemble a collection of point charges:

• For a discrete distribution: W = ½ Σ_i (q_iV(r_i))
• For a continuous distribution: W = ½ §(pV)dt

Answer 9

6

Question 9

Faraday’s Law for a moving loop (using the Lorentz force law)

Answer 9

14

Question 11

The wave equation (from Maxwell’s equations in vaccuum)

Answer 11

20

Question 12

Multipole expansion of the magnetic vector potential

Answer 12

11

Question 12

Prove that:

A_dip(r_i) = µ/(4π) · 1/r² · (m x r^{^})

B_dip(r_i) = µ/(4π) · 1/r³ · ((3(m x r^{^})r^{^} - m)

Given: §(c·r’)dl’ = -cxa

Answer 12

12

Question 13

Prove that

€/2 §(E²)dt = ½ §(pV)dt

Answer 13

7

Question 14

Prove that

1/2µ §(B²)dt = ½ M₁₁I₁² + ½ M₂₂I₂² + M₁₂I₁I₂

Answer 14

16

Question 16

Multipole expansion of the electric potential in powers of 1/r

Answer 16

8

Question 17

Prove that:

V_dip(r_i) = 1/(4π€) · 1/r² · (p·r^{^})

E_dip(r_i) = 1/(4π€) · 1/r³ · ((3(p·r^{^})r^{^} - p)

Answer 17

9

Question 18

Conservation of electric charge (from Maxwell’s equations in INTEGRAL form)

Answer 18

19

Question 19

Faraday’s law for a fixed loop (from Maxwell’s equations)

Answer 19

13

Question 20

Poynting’s Theorem (from Maxwell’s equations and the Lorentz Force Law)

Answer 20

4