Magnetic field and Induced Electric field inside a solenoid

[m0t0xk1d]

Homework Statement

The problem give is: A solenoid has N=500 windings, radius a=.1 m and a height h = .6m; the current is found to be decreasing according to I(t) = Io - bt, where Io = .4 amps and b = .2 amps/second.
Calculate the rate at which electromagnetic energy is leaving the solenoid at t=1 second. Answer this overarching question by answering the following set of guided questions.

1. using amperes law derive an expression for the magnetic field inside the solenoid. your expression for the magnetic field with be a function of time.
2. use Faraday's law to calculate the electric field inside the solenoid

Homework Equations

Faradays law = ∫E*dl = -d$\Phi$ / dt
amperes law = ∫B*dl = $\mu$o * I

The Attempt at a Solution

my questions are for part 1 and 2 not the actually over hanging question of the energy.

For part 1 my attempt was amperes law = ∫B*dl = $\mu$o * I. so B*L = $\mu$ * I. Then B = ( $\mu$ * I *N)/L. So once I got to here, to make B a function of time (B(t)) I plugged in the equation for I(t) for I and got B(t) = ( $\mu$ * N / L ) (Io - bt). I have no way to see if this is right or wrong so I wanted to see if someone could check my work.


So for part 2 I got ∫E*dl = -d$\Phi$ / dt = ∫E*dl = A dB/dt. then E *2$\pi$r = A dB/dt. Then doing some algebra I got E = R/2 *db/dt = r/2 d/dt($\mu$_oNI/L) from there on I went on to E = r/2 * $\mu$_o N/L * dI/dt. This is where it really stumps me. I don't know what to do with dI/dt. my original equation for I = Io - bt. My first thought was take the derivative which I came dI/dt = 1 - bt, but then I came to thinking b is .2 amps/second, which is already the rate at which I is decreasing, So I thought dI/dt = -b. but either was I was still unsure and looking to see if someone could help me out on this

 

[paisiello2]

Part 1 looks correct to me.

Part 2, the formula for E looks correct but dI/dt is the rate of change of I with respect to time. So you have to take the first derivative of I. I think looking at the units as you started to do will confirm the answer for you.