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.
Induced Electromotive Force (emf) and Induced Current Direction
Explanation:
An emf is induced in a conducting loop when the magnetic flux through it changes, such as when the loop enters or exits a magnetic field \( \vec{B} \). This phenomenon is described by Faraday’s Law of Induction.
Faraday's law is mathematically represented as:
\[ \mathcal{E} = - \frac{d\Phi_B}{dt} = B \cdot \frac{dA}{dt} \]
Step-by-Step Analysis:
1. Loop Entering Magnetic Field:
The area of the loop within the magnetic field increases over time.
Let \( x(t) \) be the length of the loop's side within the magnetic field at time \( t \). The magnetic flux \( \Phi_B \) is given by:
Here, \( v \) represents the velocity at which the loop moves into the magnetic field.
2. Loop Fully Within Magnetic Field:
The area within the field remains constant.
Consequently, there is no change in flux, and the induced emf is:
3. Loop Exiting Magnetic Field:
The area of the loop within the magnetic field decreases over time.
Induced Current Direction:
The direction of the induced current is governed by Lenz’s Law. This law posits that the induced current opposes the alteration in magnetic flux that produced it.
When entering, the inward magnetic flux (× direction) intensifies. Thus, the induced current flows clockwise.
When exiting, the inward magnetic flux weakens. Therefore, the induced current flows anticlockwise.
Summary:
The induced emf is determined by the rate of change of magnetic flux resulting from the loop's movement within the magnetic field. The direction of the induced current, whether entering or exiting the field, is dictated by Lenz’s Law.
