Question

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}

• A.
• B.
• C.
• D.

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.

Answer 1

The Correct Option is D

Step 1: Understanding Faraday's Law. According to Faraday's Law of Electromagnetic Induction, the induced emf \( \varepsilon \) in a loop is given by: \[ \varepsilon = \left| \frac{d\Phi_B}{dt} \right|, \] where \( \Phi_B \) is the magnetic flux through the loop. Step 2: Expressing flux in terms of motion. Since the loop is moving with constant velocity \( v \), the flux linkage \( \Phi_B \) is proportional to the area of the loop inside the magnetic field: \[ \Phi_B = B L x, \] where: - \( B \) is the magnetic field strength, - \( L \) is the width of the loop, - \( x \) is the portion of the loop still inside the field, given by \( x = vt \). Step 3: Computing emf. Differentiating \( \Phi_B \) with respect to time: \[ \varepsilon = B L \frac{dx}{dt} = B L v. \] Since \( v \) is constant, the emf remains constant while the loop is partially inside the field. However, as the loop starts exiting, the effective area inside the field decreases linearly, causing \( \varepsilon \) to decrease linearly to zero. Step 4: Identifying the correct graph. - Since the emf starts at zero, increases linearly while exiting, and reaches a peak before going to zero once the loop is fully out of the field, the correct choice is: \[ \boxed{1} \text{ (Linearly increasing graph)} \]

Answer 2

Step 1: Understanding the situation. 
Let the uniform magnetic field \( B \) be directed into the plane of the loop. The rectangular loop of length \( l \) is moving with a constant speed \( v \) out of the field region.

Step 2: Expression for induced emf.
When a conducting loop moves in or out of a magnetic field, the induced emf is given by: \[ \varepsilon = B l v \] but only during the time interval when the loop is cutting magnetic field lines — that is, while it is entering or leaving the field region.

Step 3: Time variation of emf.
- When the loop is completely inside the field, no change in flux occurs → \( \varepsilon = 0 \). - When the loop is completely outside the field, again no flux → \( \varepsilon = 0 \). - When it is partially inside, magnetic flux through it changes uniformly with time → induced emf is constant in magnitude.

Step 4: Nature of induced emf with time.
While entering the field → induced emf has one polarity (say, positive). While leaving the field → induced emf has the opposite polarity (negative). However, the question asks for the magnitude of emf, so polarity is ignored.

Therefore, the magnitude of emf:

• is zero when the loop is completely inside or outside,
• is constant when partially entering or leaving,
• shows two equal rectangular pulses separated by a zero region in between.

 

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Step 5: Sketch / Conclusion.

Hence, the graph of \( |\varepsilon| \) vs \( t \) will consist of two equal rectangular pulses, indicating constant induced emf during entry and exit from the field region.

✅ Correct graph: Option (4)