A region in the form of an equilateral triangle (in x-y plane) of height L has a uniform magnetic field π΅β pointing in the +z-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height πΏ, is placed in the x-y plane with its vertex P at x = 0 in the orientation shown in the figure. At π‘ = 0, the loop starts entering the region of the magnetic field with a uniform velocity π£ along the +x-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graph best depicts the variation of the induced emf (E) in the loop as a function of the distance (π₯) starting from π₯ = 0?
Which of the following graph best depicts the variation of the induced emf (E) in the loop as a function of the distance (π₯) starting from π₯ = 0?
The Correct Option is A
Approach Solution - 1
To solve the problem, we need to analyze the variation of the induced emf in a conducting triangular loop moving through a region of uniform magnetic field.
1. Understanding the Setup:
- The loop PQR is an equilateral triangle of height \( L \) moving along the \( +x \)-direction.
- The magnetic field \( \vec{B} \) is uniform and points in the \( +z \)-direction.
- At \( x=0 \), vertex P enters the magnetic field region.
- The induced emf \( E \) is related to the rate of change of magnetic flux through the loop.
2. Variation of Flux and emf:
- Initially, no part of the loop is inside the magnetic field, so \( E = 0 \).
- As the loop enters the field (from \( x=0 \) to \( x=L \)), the area inside the field increases linearly, so flux increases linearly.
- The emf \( E = -d\Phi/dt \) is proportional to the rate of change of flux.
- When half the loop enters (\( x=L/2 \)), the rate of change of area changes slope.
- Between \( x=L \) and \( x=2L \), the loop is leaving the field and the flux decreases.
- Beyond \( x=2L \), the loop is completely outside, so \( E = 0 \).
3. Shape of emf vs. distance graph:
- The emf starts at zero at \( x=0 \), decreases (negative slope) until \( x=L/2 \), then increases until \( x=L \).
- After \( x=L \), emf is zero until \( x=3L/2 \), then increases positively until \( x=2L \), and finally returns to zero.
- This corresponds to graph (A) with negative and positive triangular peaks.
Final Answer:
Option (A)
Approach Solution -2
To solve the problem, analyze how the induced emf in the loop changes as the loop moves through the magnetic field region.
1. Setup:
- The loop PQR is an equilateral triangle of height \(L\), moving along the +x direction.
- Magnetic field \(\vec{B}\) is uniform and directed along +z.
- Induced emf \(E\) is related to the rate of change of magnetic flux through the loop.
- At \(x=0\), vertex P enters the magnetic field region.
2. Flux and emf behavior:
- For \(0 \leq x < L\): increasing part of the loop enters the field.
- The flux increases non-linearly, first faster then slower due to triangular shape.
- This results in a changing induced emf with negative and positive slopes at different points.
- For \(L \leq x \leq 2L\): the loop leaves the magnetic field region.
- The flux decreases similarly, giving a characteristic emf pattern.
3. Graph Interpretation:
- The induced emf first decreases (negative peak) as the loop enters.
- Then it increases to zero at \(x=L\).
- After that, it stays zero for some region.
- Finally, it increases positively as the loop leaves the field and goes back to zero at \(x=2L\).
- This matches graph (A).
Final Answer:
Option (A)
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