A coil and a magnet moves with their constant speeds `5m//sec` and `3m//sec`. Respectively , towards each other, then induced emf in coil is 16 m V . If both are moves in same direction , then induced emf in coil :-

Options

A. 15mV
B. 4m V
C. 64m V
D. Zero

Answer

To solve the problem step by step, we can follow this procedure:

### Step 1: Understand the Problem
We have a coil and a magnet moving towards each other with speeds of 5 m/s and 3 m/s, respectively. The induced EMF (electromotive force) in the coil when they are moving towards each other is given as 16 mV. We need to find the induced EMF when both the coil and the magnet are moving in the same direction.

### Step 2: Calculate the Relative Velocity When Moving Towards Each Other
When the coil and the magnet move towards each other, the relative velocity (v_cm) is the sum of their speeds:
\[ v_{cm} = v_c + v_m = 5 \, \text{m/s} + 3 \, \text{m/s} = 8 \, \text{m/s} \]

### Step 3: Establish the Relationship Between EMF and Relative Velocity
According to Faraday's law of electromagnetic induction, the induced EMF (E) is directly proportional to the relative velocity:
\[ E \propto v_{cm} \]
This means we can express the induced EMF as:
\[ E = k \cdot v_{cm} \]
where \( k \) is a proportionality constant.

### Step 4: Write the Equation for the Induced EMF in the First Case
From the information given:
\[ E = 16 \, \text{mV} \quad \text{when} \quad v_{cm} = 8 \, \text{m/s} \]
Thus, we can write:
\[ 16 \, \text{mV} = k \cdot 8 \, \text{m/s} \]

### Step 5: Calculate the Proportionality Constant (k)
From the equation above, we can solve for \( k \):
\[ k = \frac{16 \, \text{mV}}{8 \, \text{m/s}} = 2 \, \text{mV/(m/s)} \]

### Step 6: Calculate the Relative Velocity When Moving in the Same Direction
When both the coil and the magnet move in the same direction, the relative velocity (v_cm') is given by:
\[ v_{cm}' = v_c - v_m = 5 \, \text{m/s} - 3 \, \text{m/s} = 2 \, \text{m/s} \]

### Step 7: Write the Equation for the Induced EMF in the Second Case
Using the same proportionality constant \( k \):
\[ E' = k \cdot v_{cm}' \]
Substituting the values:
\[ E' = 2 \, \text{mV/(m/s)} \cdot 2 \, \text{m/s} = 4 \, \text{mV} \]

### Step 8: Conclusion
The induced EMF in the coil when both the coil and the magnet are moving in the same direction is:
\[ E' = 4 \, \text{mV} \]