# Solving the Emf Problem: Induced EMF Calculation with Faraday's Law

• robert25pl
In summary, you are trying to find the time rate of change of flux through a loop by integrating over the area. You need to integrate over x only because B is a function of x only.
robert25pl
A magnetic field is given in the xz-plane by B=Bo/x j Wb/M^2. Consider a rigid square loop situated in the xz-plane with its vertices at (Xo,Zo), (Xo,Zo+b),(Xo+a,Zo+b) and (Xo+a,Zo). If the loop is moving in that plane with the velocity $$V = V_{o}\vec{i}$$ m/s what is the induced emf using Faraday's law

Can someone check my magnetic flux set up

$$\psi=\int_{s}B\cdot\,ds=\int_{x=0}^{x+Vot} \int_{z=0}^{b}\frac{Bo}{x}\vec{j}\cdot\, dx\,dz\vec{j}$$

First find the position of the corners of the loop for any time. Remember, both sides of the loop are moving in the x direction (right now you only have one moving, which means your loop is expanding). Don't forget the corners of the loop are at x0 and z0, not 0 and 0.

$$\psi=\int_{s}B\cdot\,ds=\int_{xo}^{xo+Vot} \int_{zo}^{zo+Vot}\frac{Bo}{x}\vec{j}\cdot\, dx\,dz\vec{j}$$

My problem is that I don't understand well how to get the position of the corners of the loop for any time because there is not even one example with moving loop in the book.

Last edited:
robert25pl said:
A magnetic field is given in the xz-plane by B=Bo/x j Wb/M^2. Consider a rigid square loop situated in the xz-plane with its vertices at (Xo,Zo), (Xo,Zo+b),(Xo+a,Zo+b) and (Xo+a,Zo). If the loop is moving in that plane with the velocity $$V = V_{o}\vec{i}$$ m/s what is the induced emf using Faraday's law

Can someone check my magnetic flux set up

$$\psi=\int_{s}B\cdot\,ds=\int_{x=0}^{x+Vot} \int_{z=0}^{b}\frac{Bo}{x}\vec{j}\cdot\, dx\,dz\vec{j}$$
You want to find the time rate of change of flux so:

Emf $$= \frac{d\phi}{dt} =\frac{d}{dt}\int_{A}B\cdot dA$$

To express flux as a function of t:

$$\phi(x) = \int_{x}^{x + a} b \frac{B_0}{x}dx = bB_0\int_{x}^{x + a} \frac{1}{x}dx = bB_0(ln(\frac{x+a}{x}))$$

Since x = vt:

$$\phi(t) = bB_0(ln(\frac{vt+a}{vt}))$$

So the time rate of change of flux is?...

AM

So my approach is wrong?

$$\psi=\int_{s}B\cdot\,ds=\int_{xo}^{xo+Vot} \int_{zo}^{zo+Vot}\frac{Bo}{x}\vec{j}\cdot\, dx\,dz\vec{j}$$

and then find emf

$$\oint_{c}E\cdot\,dl=-\frac{d}{dt} \int_{s}B\cdot dS$$

I know my set up of position of corners at any time could be wrong, but why didn't you integrate wrt dz. I understand that the magnetic field is independent of z. Thanks

robert25pl said:
So my approach is wrong?

$$\psi=\int_{s}B\cdot\,ds=\int_{xo}^{xo+Vot} \int_{zo}^{zo+Vot}\frac{Bo}{x}\vec{j}\cdot\, dx\,dz\vec{j}$$
Just a little confused.

and then find emf

$$\oint_{c}E\cdot\,dl= -\frac{d}{dt} \int_{s}B\cdot dS$$

I know my set up of position of corners at any time could be wrong, but why didn't you integrate wrt dz. I understand that the magnetic field is independent of z.
The left side is just the electric potential or emf. The right side is the rate of change of flux (I prefer to use A for area to avoid confusion with Ampere's law.)

To work out the flux through the loop, you have to integrate B over the area. But B is a function of x only, so you can avoid integrating over z by simply letting dA = bdx and integrating over x only. Dividing the area into little strips of length b and width dx and integrating B from x = x to x = x+a gives you the
total flux at a given point.

AM

Now I understood very well. Thank you very much.

## 1. What is Faraday's Law?

Faraday's Law is a fundamental law in electromagnetism that states that the magnitude of the induced electromotive force (EMF) in a closed circuit is equal to the rate of change of the magnetic flux through the circuit.

## 2. Why is it important to solve the EMF problem?

The EMF problem refers to the calculation of induced EMF in a circuit, which is crucial in understanding the behavior of electromagnetic devices and circuits. It allows us to predict and control the flow of electric current and design efficient devices.

## 3. How is EMF calculated using Faraday's Law?

EMF is calculated by multiplying the rate of change of magnetic flux with the number of turns in the circuit. Mathematically, it can be expressed as EMF = -NΔΦ/Δt, where N is the number of turns and ΔΦ/Δt is the change in magnetic flux over time.

## 4. What are some practical applications of Faraday's Law and solving the EMF problem?

Some practical applications include generators, transformers, electric motors, and induction cooktops. These devices use Faraday's Law and the concept of induced EMF to convert mechanical energy into electrical energy or vice versa.

## 5. Are there any limitations to Faraday's Law and solving the EMF problem?

While Faraday's Law is a fundamental principle in electromagnetism, it does have some limitations. It assumes ideal conditions and does not take into account factors such as resistance, capacitance, and inductance. Additionally, the calculations can become more complex in cases of non-uniform magnetic fields or changing circuit geometries.

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