Induced electric field in a solenoid

[richyw]

Homework Statement

A long solenoid with 1000 turns per meter and radius 2cm carries an oscillating current given by (5A)\sin(100\pi t). What is the electric field induced at radius r=1cm from the axis of the solenoid? What is the direction of this electric field when the current is increasing counterclockwise in the coil?

Homework Equations

\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt}
\Phi_B=\oint\vec{B}\vec{dA}and I have this on my formula sheet so I will start at this point (magnetic field inside a solenoid with n turns per unit length and current IB=\mu_0nI

The Attempt at a Solution

So I think in this situation I can say \Phi_B=BA=(\mu_0nI)(\pi r^2)So\frac{d\Phi_B}{dt}=\pi\mu_0nr^2\frac{dI}{dt}
Also
\oint\vec{E}\cdot d\vec{l}=2E\pi r
So I got (ignoring the negative, just considering the magnitude)
E=\frac{\pi\mu_0nr^2}{2\pi r}\frac{dI}{dt}= \frac{1}{2} \mu_0nr\frac{dI}{dt}=\frac{1}{2}\mu_0nr\omega I_{max}\cos(\omega t)where \omega=100\pi and I_{max}=5A

This makes sense to me but is not getting me the correct answer! Also I only reasoned that the electric field would be clockwise, but not really sure on that part. Does anyone see where I am going wrong?

 

[TSny]

Your work looks good to me. Did you convert r to SI units?

 

[richyw]

ah sorry. this was the correct answer! the solution I was looking at messed up. what a waste of time typing that out! thank you for looking at this for me!