# How to calculate the Induced Voltage Given Spinning Conducting Loop? Intro to Electromagnetics problem

I'm having trouble figuring out a different variation of this problem. On my homework, I'm asked to find the induced EMF for the following B field: $$\vec{B}=50\vec{a_y}$$

I was able to figure out the problem with the B field being: $$\vec{B}=50\vec{a_x}$$

I want to know how does my equations change if the B field is in the $\vec{a_y}$ direction. I'm confused shouldn't it be the same due to symmetry :) The back of my book says I'm right for $$\vec{B}=50\vec{a_x}$$ but not for $$\vec{B}=50\vec{a_y}$$

The following is my solution to the problem if $$\vec{B}=50\vec{a_x}$$

Question:

The loop shown in Figure 9.7 is inside a uniform magnetic field $\vec{B}=50\vec{a_x}$ mWB/$m^2$. If side DC of the loop cuts the flux lines at the frequency 50Hz and the loop lies in the yz plane.

Find the following:

a)The induced EMF at $t=1$ms

Figure 9.7

Let's use the general form of Faraday's Law.

$$V_{emf} = -\frac{d}{dt} \int_S \!\vec{B}\cdot\vec{dS}$$

1) We can simplify the integral as $$\int_S \!\vec{B}\cdot\vec{dS}=B\cos{(\phi)}zy$$ where z and y is simply the length and width of the square conducting loop.

2) Now we have the following expression to evaluate: $$V_{emf} = -\frac{d}{dt} B\cos{(\phi)}zy$$

3) We can find $\phi$ by noting $\omega=2\pi f$ so $\phi = \omega t$

So... $$V_{emf} = -\frac{d}{dt} B\cos{(\omega t)}zy$$

4) Now evaluate the derivative: $$V_{emf} = -Bzy \frac{d}{dt}\cos{(\omega t)}$$ $$V_{emf} = Bzy\omega\sin{(\omega t)}$$

5) Finally plug in the known numbers to find the $V_{emf}$. at $t=1ms$

$$z=3*10^{-2}$$ $$y=4*10^{-2}$$ $$f=50Hz --> \omega = 2\pi f = 100\pi$$ $$t=1ms$$ $$B=(50*10^{-3})$$

$$V_{emf} = 100\pi(50*10^{-3})(3*10^{-2})(4*10^{-2})\sin{(100\pi (1*10^{-3}))}$$

Therefore: $$V_{emf} = 5.825mV$$

• +1 - Excellent use of MathJax. Here at EE.SE, the delimiter for inline MathJax is $ instead of the more popular . I've replaced those for you. Please check them. I hope I didn't add any errors to the expressions. Sep 17, 2014 at 1:48 • Cross posted at physics.SE thus explaining the  in place of$ : physics.stackexchange.com/q/135859/9887 Sep 17, 2014 at 1:53
• Alfred Centauri, I thought of posting it on both forums since it is relevant to either. I'm sorry is that a bad thing I did? I'm new to both forums. Sep 17, 2014 at 2:05
• Ricardo, thanks! I just learned LaTeX today. Somebody at the Physics forum told me to learn it because taking a picture of my equations is to messy. haha. Thanks for the uplift. :) Thanks, I think everything is intact. Also, how do I specifically quote people so they know I responded to their comment? Is a name enough or there is someway to tag them? Sep 17, 2014 at 2:06
• You're welcome, @TwilightSparkleTheGeek. About cross-posting: isn't very well accepted on StackExchange. Here's more info on that. A more accepted approach is to ask the question in one stack and if you don't get the answer you want, you either ask a moderator to migrate it, or delete and ask it in the other stack. Sep 17, 2014 at 3:10