A uniform magnetic field existsin region given by `vec(B) = 3 hat(i) + 4 hat(j)+5hat(k)`. A rod of length `5 m` is placed along `y`-axis is moved along `x`- axis with constant speed `1 m//sec`. Then the magnitude of induced `e.m.f` in the rod is :

Options

A. (a) `0`
B. (B) `25 V`
C. ( c) `20V`
D. (d) `15 V`

Explanation

To find the magnitude of the induced electromotive force (e.m.f) in the rod, we can follow these steps:

### Step 1: Identify the Given Values
- Magnetic field \( \vec{B} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \) (in Tesla)
- Length of the rod \( L = 5 \, \text{m} \) (along the y-axis)
- Velocity of the rod \( \vec{V} = 1 \, \text{m/s} \) (along the x-axis)

### Step 2: Write the Vectors
- The velocity vector of the rod can be expressed as \( \vec{V} = 1 \hat{i} \, \text{m/s} \).
- The length vector of the rod, which is along the y-axis, can be expressed as \( \vec{L} = 5 \hat{j} \, \text{m} \).

### Step 3: Use the Formula for Induced EMF
The induced e.m.f \( E \) can be calculated using the formula:
\[ E = |\vec{V} \times \vec{B} \cdot \vec{L}| \]

### Step 4: Calculate the Cross Product \( \vec{V} \times \vec{B} \)
To find \( \vec{V} \times \vec{B} \):
\[ \vec{V} \times \vec{B} = (1 \hat{i}) \times (3 \hat{i} + 4 \hat{j} + 5 \hat{k}) \]
Using the properties of the cross product:
- \( \hat{i} \times \hat{i} = 0 \)
- \( \hat{i} \times \hat{j} = \hat{k} \)
- \( \hat{i} \times \hat{k} = -\hat{j} \)

Calculating the components:
\[ \vec{V} \times \vec{B} = 1 \hat{i} \times (3 \hat{i}) + 1 \hat{i} \times (4 \hat{j}) + 1 \hat{i} \times (5 \hat{k}) \]
\[ = 0 + 4 \hat{k} - 5 \hat{j} \]
So,
\[ \vec{V} \times \vec{B} = -5 \hat{j} + 4 \hat{k} \]

### Step 5: Calculate the Dot Product with \( \vec{L} \)
Now, we need to calculate the dot product:
\[ (\vec{V} \times \vec{B}) \cdot \vec{L} = (-5 \hat{j} + 4 \hat{k}) \cdot (5 \hat{j}) \]
Calculating the dot product:
\[ = (-5)(5) + (4)(0) = -25 \]

### Step 6: Find the Magnitude of Induced EMF
The magnitude of the induced e.m.f is:
\[ E = |-25| = 25 \, \text{V} \]

### Conclusion
The magnitude of the induced e.m.f in the rod is \( 25 \, \text{V} \).

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