# Applying amperes law and faradays law to a solenoid and energy density

• jb646
In summary, the conversation discussed how to calculate the magnetic field, induced electric field, and energy density inside a solenoid with a given current and dimensions using Ampere's law, Faraday's law, and the equation for energy density. The equations were provided and simplified for each part.
jb646
This isn't really a homework problem, I just need to know how to do a problem similar to this one for the final and I don't want to fail, so I posted it here.

A solenoid with n turns per unit length, length l and radius a (l>>a) has a current I(t)==Io sin(wt) [w is omega], where I and w are constant.
a) using ampere's law calculate the magnetic field inside the solenoid.
b) use faraday's law to calculate the induced electric field around the edge of the solenoid
c) calculate the energy density due to the electric and magnetic field inside the solenoid.

BL=uNI
∇xE= -(dB/dt) <--partial derivative of mag field
U=(εo/2)*E^2+(1/2*uo)*B^2

a) For the first part I know that BL=uNI so I tried just getting B=u*n(Iosin(wt)) but there must be more to it.
b) so if that is true do I just stick that into equation 2 and solve for E
c) and if both of those are right, do I just stick them in equation 3 or is there more I have to do for that one too?

For part a) you need to use Ampere's law: B = μo*(N*I/l). So B = μo*(n*Io*sin(ωt)/l). For part b) you can then use Faraday's law: ∇xE = - (dB/dt). Plugging in the equation for B, you get ∇xE = - μo*(n*Io*ω*cos(ωt)/l).For part c) you can then use the equation for energy density: U = (εo/2)*E^2 + (1/2*μo)*B^2. Plugging in the equations for E and B, you get U = (εo/2)*[μo*(n*Io*ω*cos(ωt)/l)]^2 + (1/2*μo)*[μo*(n*Io*sin(ωt)/l)]^2. Simplifying this gives U = (εo*μo*n^2*Io^2*ω^2*cos(ωt)^2 + μo^2*n^2*Io^2*sin(ωt)^2)/(2*l^2).

## What is Amperes law?

Amperes law is a fundamental law in electromagnetism that relates the magnetic field produced by a current-carrying wire to the current enclosed by a closed loop around that wire.

Faradays law is a fundamental law in electromagnetism that states that a changing magnetic flux through a surface will induce an electric field around that surface.

## How is Amperes law applied to a solenoid?

Amperes law can be applied to a solenoid by considering the current enclosed by a closed loop around the solenoid. The magnetic field produced by this current can then be determined using Amperes law.

## How is Faradays law applied to a solenoid?

Faradays law can be applied to a solenoid by considering the changing magnetic flux through the surface of the solenoid. This changing flux can induce an electric field within the solenoid according to Faradays law.

## What is the energy density of a solenoid?

The energy density of a solenoid is the amount of energy stored per unit volume within the solenoid. It can be calculated by taking the product of the magnetic field strength and the square of the number of turns per unit length within the solenoid.

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