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 I$$B=\mu_0nI$$

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!