Question

a) Suppose that the current in the solenoid is I(t). Within the solenoid, but far from its ends, what is the magnetic field B(t) due to this current? b) What is the magnetic flux Phi_1(t) through a single turn of the solenoid? c) Suppose that the current varies with time, so that dI(t)/dt neq 0. Find the electromotive force EMF induced across the entire solenoid due to the change in current through the entire solenoid. d) The self-inductance Lis related to the self-induced EMF EMF(t) by the equation {cal{E}}(t) = - L dI(t)/dt. Find Lfor a long solenoid. (Hint: The self-inductance L will always be a positive quantity.)

Answer 1

Answer:

please see the answers below

Explanation:

a) the magnetic field is given by

$B=\frac{\mu_0NI(t)}{l}$

N is the number of turns, l is the length of the solenoid, mu_0 is the magnetic permeability of vacuum and I(t) is the current.

b)

$\Phi_B=BS=\frac{\pi r^2 \mu_0 N I(t)}{l}$

for a single turn:

$\Phi_B=BS=\frac{\pi r^2 \mu_0I(t)}{l}$

c)

$emf=-\frac{d\Phi_B}{dt}=-\frac{\pi r^2 \mu_0 N}{l}\frac{dI(t)}{dt}$

d)

$emf=-L\frac{dI(t)}{dt}\\\\L=-\frac{emf}{\frac{dI(t)}{dt}}=-\frac{-\frac{\pi r^2\mu_0N}{l}\frac{dI(t)}{dt}}{\frac{dI(t)}{dt}}=\frac{\pi r^2 \mu_0N}{l}$

hope this helps!!