# Motional EMF through a circular magnetic field

1. Feb 28, 2010

### Sancor

1. The problem statement, all variables and given/known data

A circuit consists of three fixed sides shaped like a |_| and a sliding rod of length 2R. The rod slides at constant speed v into a region of nonzero field B coming perpendicularly out of the paper and limited to a circle of diameter 2R. What is the magnitude and sense (clockwise or anti) of the emf in the circuit as a function of time, with t=0 the time the rod first hits the B field? Hint: choose your origin to be the point of first contact of rod with B field and write an equation for the circle.

Here is my attempt at a figure to better visualize this for those who might help

|O|
|=|
|_|

where | are fixes rails, O is the magnetic field w/ radius R and = is the moving rod of length 2R

2. Relevant equations
Emf = -$$\frac{d\Phi}{dt}$$
$$\Phi$$=$$\int$$B dA

3. The attempt at a solution

So it's clear that the induced emf is induced in order to resist the increasing flux out of the page, so its sense will be clockwise. and the necesarry equations for figuring out the emf is faraday's law of induction. Now what I can't get to work is a mathematical expresion for all of this. When I try integrating the equation of the circle with the origin at the point of first intersection in respect to y from 0 to 2R (to check that it gives $$\pi$$R2) it gives me a residual sine(infinity) term.

x= $$\sqrt{}2yR-y^2$$

so area should be
$$\int2$$\sqrt{}2yR-y^2$$dy$$

If i treat it as if the origin is the center of the circle however i get the expected area result from the integral. But when i put this in the d/dt BA to find the emf, the expression is too complex to resolve like a typical motional emf problem where y would typically just become v.