A rectangular metallic loop is moving out of a uniform magnetic field region to a field-free region with a constant speed. When the loop is partially inside the magnetic field, the plot of the magnitude of the induced emf \( (\varepsilon) \) with time \( (t) \) is given by:
\includegraphics[width=1\linewidth]{3.png}
Show Hint
To determine the emf induced in a moving conductor:
- Use Faraday’s Law: \( \varepsilon = \left| \frac{d\Phi_B}{dt} \right| \).
- If motion is constant and uniform, the change in flux is linear, leading to a linear change in emf.
- For non-uniform motion, consider the velocity function to determine the rate of flux change.
Step 1: Understanding Faraday's Law.
Faraday's Law of Electromagnetic Induction states that the induced electromotive force (emf), denoted by \( \varepsilon \), in a loop is equal to the absolute value of the rate of change of magnetic flux \( \Phi_B \) through the loop:
\[
\varepsilon = \left| \frac{d\Phi_B}{dt} \right|,
\]
where \( \Phi_B \) represents the magnetic flux.
Step 2: Expressing flux in terms of motion.
Given that the loop moves with a constant velocity \( v \), the magnetic flux linkage \( \Phi_B \) is directly proportional to the area of the loop that is within the magnetic field. This can be expressed as:
\[
\Phi_B = B L x,
\]
with the following parameters:
- \( B \) representing the magnetic field strength.
- \( L \) representing the width of the loop.
- \( x \) representing the segment of the loop still within the magnetic field, which is defined as \( x = vt \).
Step 3: Computing emf.
By differentiating \( \Phi_B \) with respect to time, we obtain the emf:
\[
\varepsilon = B L \frac{dx}{dt} = B L v.
\]
As \( v \) is constant, the emf remains constant while the loop is partially within the field. However, when the loop begins to exit the field, the effective area inside the field diminishes linearly, causing \( \varepsilon \) to decrease linearly to zero.
Step 4: Identifying the correct graph.
- The emf originates from zero, exhibits a linear increase as the loop exits, reaches a maximum, and then returns to zero once the loop is completely outside the field. Therefore, the correct representation is:
\[
\boxed{1} \text{ (Linearly increasing graph)}
\]
