A square shaped loop of side \(\frac{l}{2}\) is initially lying outside a region of uniform magnetic field →B as shown in the figure. The loop is moved towards right with a constant velocity →v till it goes out of the region of magnetic field.
As a conducting loop enters or exits a magnetic field \( \vec{B} \), the magnetic flux through the loop changes, inducing an emf according to **Faraday’s Law of Induction**.
Faraday's law is expressed as:
\[ \mathcal{E} = - \frac{d\Phi_B}{dt} = B \cdot \frac{dA}{dt} \]
Step-by-Step Analysis:
1. When the Loop Enters the Magnetic Field:
The area inside the magnetic field increases with time.
Let the length of the loop's side inside the magnetic field be \( x(t) \) at time \( t \). Then the magnetic flux \( \Phi_B \) is:
Here, \( v \) is the velocity of the loop moving into the magnetic field.
2. When the Loop is Fully Inside the Magnetic Field:
The area inside the field becomes constant.
Hence, no change in flux occurs and the induced emf is:
3. When the Loop Exits the Magnetic Field:
The area inside the magnetic field decreases with time.
Induced Current Direction:
The direction of the induced current is determined by **Lenz’s Law**, which states that the induced current will oppose the change in flux.
While entering, the magnetic flux increases inward (× direction), so the induced current is **clockwise**.
While exiting, the magnetic flux decreases, so the induced current is **anticlockwise**.
Summary:
The induced emf is calculated based on the change in magnetic flux due to the motion of the loop through the magnetic field. The direction of the induced current depends on whether the loop is entering or exiting the magnetic field, as determined by Lenz’s Law.
