Motional EMF through a circular magnetic field

[Sancor]

Homework Statement

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

Homework Equations

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

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
[tex]\int2$$\sqrt{}2yR-y^2$$dy [/tex]

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.

Any advice?

 

[Jhero]

Hello! I would like to provide some guidance on how to approach this problem.

First, let's start by defining some variables:
- R = radius of the circle (fixed sides)
- v = velocity of the sliding rod
- t = time
- B = magnetic field
- A = area of the circle (limited by the fixed sides)

Now, let's think about the motion of the sliding rod. At t=0, the rod first hits the magnetic field. This means that the area of the circle is increasing with time, as the rod slides further into the field. We can express this mathematically as:
A = \pi(R + vt)^2

Next, we need to find the rate of change of this area with time, in order to calculate the emf. This can be done by taking the derivative of A with respect to time:
\frac{dA}{dt} = 2\pi(R + vt)v

Now, we can use Faraday's law of induction to calculate the induced emf:
Emf = -\frac{d\Phi}{dt} = -\frac{d}{dt}(\int B dA) = -\int B\frac{dA}{dt}dt = -\int B2\pi(R + vt)vdt

Since we are only interested in the magnitude of the emf, we can ignore the negative sign and solve for the absolute value:
|Emf| = \int B2\pi(R + vt)vdt = 2\pi B\int(R + vt)vdt = 2\pi B(\frac{1}{2}R^2 + \frac{1}{2}vt^2)

Now, we can see that the emf is a function of both time and the velocity of the rod. As the rod slides at a constant speed, the emf will increase with time until it reaches a maximum value when the rod is fully inside the magnetic field.

To determine the sense of the emf, we can use Lenz's law, which states that the induced emf will always oppose the change in flux. In this case, as the area of the circle increases, the flux out of the page also increases. Therefore, the induced emf will be in the clockwise direction, as you correctly predicted.

I hope this helps guide you towards finding the correct mathematical expression for the emf in this