Poynting Vector - Finding stored energy per unit length of a solenoid

physicsoxford

1. The problem statement, all variables and given/known data

long solenoid of n turns per unit length is wound upon a cylindrical core of radius a
and relative permeability. The current I through the solenoid is increasing with time t at a
constant rate. Obtain expression for the rate of increase of stored energy per unit length in the core
of the solenoid
(a) from the inductance per unit length of the solenoid, and dI=dt.
(b) from the energy associated with the fields internal to the solenoid core.
(c) by integration of the Poynting vector over an appropriate surface.

2. Relevant equations

None are given but I believe this is what should be considered:

-∂/∂t ∫[ εE²/2 + B²/2μ ] dv = ∫ J_f dot E dv + ∫ E cross H da

Ampere's Law

E cross H = S

3. The attempt at a solution

Using Ampere's Law: B=μnI

L = flux/I
L = μn²lA Where l is some length and A is a surface


Part A)

∅ = L dI/dt

∅ = μn²lA dI/dt

U = Q∅/2

U/dt = μn²lA (dI/dt) (Q/dt) (1/2)

U/dt = μn²lA (dI/dt) I (1/2)

This just does not seem right to me??

Part B)

U = ∫B²/2μ dv

U = ∫(μnI)²/2μ dv

U = μn²lAI (1/2)

U/dt = μn²lA (dI/dt) I (1/2)

Part C)

U = ∫ S dv

where S = E cross H, assuming ∫ J_f dot E dv = 0

This is where I am confused. Is there an electric field in the solenoid? If so then did I not do the other parts correctly? What am I missing here...

rude man

Not that complicated. Given inductance L, what is the formula for stored energy?

Then, calculate L per unit length.