# A Simple Way to Measure Magnetic Fields

1. Homework Statement

A loop of wire is at the edge of a region of space containing a uniform magnetic field B. The plane of the loop is perpendicular to the magnetic field. Now the loop is pulled out of this region in such a way that the area A of the coil inside the magnetic field region is decreasing at the constant rate . That is, $$\frac{dA}{dt} = -c$$ , with c >0 .

The induced emf in the loop is measured to be V. What is the magnitude B of the magnetic field that the loop was in?
Express your answer in terms of some or all of the variables A ,c , and V.

2. Homework Equations

$$\epsilon = | \frac{d\Phi_m}{dt} |$$
$$\Phi_m = AB$$
$$\epsilon = lvB$$

3. The Attempt at a Solution

I get these three hints:

Hint 1. The formula for the magnetic flux through a loop

Hint 2. How to take the derivative of the product of two functions

Hint 3. The formula for the emf induced in a loop (Faraday's law)

So I know $$B = \frac{\Phi_m}{A}$$, would I just find the derivative of that?

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rock.freak667
Homework Helper
ok well you know $\Phi=BA$

and Faraday's law is that the emf induced.$E=-\frac{-d\Phi}{dt}$
so

$$E= - \frac{d\Phi}{dt} = - \frac{d}{dt}(BA)$$

B is constant so you can can remove it from inside the brackets...Can you see it better now?

So it's like this:

$$\Phi = BA$$

$$- \frac{d\Phi }{dt} =-B \frac{dA}{dt}$$

$$E = -B(-c) = Bc$$

which implies that B = E/c (assuming I didnt make any mistakes), but it says I need to express it in terms of A, c, or V.

Did I make a mistake anywhere?

rock.freak667
Homework Helper
Well E is really V..so V=Bc.

So for the second part of the question, I have to find the value of c and terms of v and L

$$c = \frac{V}{B} = \frac{vLB}{B} = vL$$

I didnt make any stupid mistakes, did I?

rock.freak667
Homework Helper
Or you could have done it in a different way and say that in 1s the coil moves vm so that the area swept out in 1s is vLm^2. meaning that dA/dt=-c=vL.