Consider a square loop of edge a moving through a uniform magnetic field at a constant velocity v. What is the induced emf?
Answer
Induced EMF in a Square Loop Moving Through a Magnetic Field
When a square loop of side length a moves with constant velocity v through a uniform magnetic field B, perpendicular to its plane, an electromotive force (EMF) is induced due to the change in magnetic flux through the loop.
🔹 Step 1: Magnetic Flux Calculation
The magnetic flux through the area of the loop inside the field is given by:
Since the loop moves at speed v and enters the field, the distance moved into the field is:
Then, the area inside the field becomes:
So, the magnetic flux is:
🔹 Step 2: Apply Faraday’s Law
Faraday’s Law states:
Substitute the flux expression and differentiate:
Hence, the induced EMF is:
The negative sign indicates the direction of induced EMF opposes the change (Lenz’s Law). For magnitude:
📘 Final Expression
The magnitude of the EMF induced in the square loop as it enters or exits the magnetic field is:
📊 Summary Table
| Symbol | Description | Unit |
|---|---|---|
| a | Length of square loop’s side | meters (m) |
| v | Velocity of the loop | m/s |
| B | Magnetic field strength | Tesla (T) |
| ΦB | Magnetic flux | Weber (Wb) |
| ε | Induced EMF | Volts (V) |
✅ Conclusion
As the square loop moves through the magnetic field, the change in flux induces an EMF that is directly proportional to the magnetic field strength (B), the side length of the loop (a), and the speed of motion (v).