A metallic rod of length 20 cm is placed in North - South direction and is moved at a constant speed of 20m/s toward East. The horizontal component of the Earth's magnetic field at the place is ` 4 xx 10 ^(-3)` T and the angle of dip is `45^(@)`. The emf induced in the rod is _______mV.

Explanation

To solve the problem of finding the induced EMF in a metallic rod moving in a magnetic field, we can follow these steps:

### Step 1: Understand the given parameters
- Length of the rod (L) = 20 cm = 0.2 m
- Speed of the rod (v) = 20 m/s
- Horizontal component of Earth's magnetic field (B_H) = \(4 \times 10^{-3}\) T
- Angle of dip (θ) = 45°

### Step 2: Determine the vertical component of the magnetic field
Since the angle of dip is 45°, the vertical component (B_V) of the Earth's magnetic field is equal to the horizontal component:
\[ B_V = B_H = 4 \times 10^{-3} \, \text{T} \]

### Step 3: Use the formula for induced EMF
The induced EMF (ε) in a rod moving in a magnetic field can be calculated using the formula:
\[ \epsilon = L \cdot v \cdot B \cdot \sin(\theta) \]
Where:
- L = length of the rod
- v = velocity of the rod
- B = vertical component of the magnetic field
- θ = angle between the velocity vector and the magnetic field vector

### Step 4: Identify the angle between velocity and magnetic field
Since the rod is moving horizontally (to the east) and the vertical component of the magnetic field is directed vertically downwards, the angle (θ) between the velocity of the rod and the vertical magnetic field is 90°:
\[ \sin(90^\circ) = 1 \]

### Step 5: Substitute the values into the formula
Now we can substitute the values into the EMF formula:
\[ \epsilon = L \cdot v \cdot B_V \cdot \sin(90^\circ) \]
\[ \epsilon = 0.2 \, \text{m} \cdot 20 \, \text{m/s} \cdot (4 \times 10^{-3} \, \text{T}) \cdot 1 \]

### Step 6: Calculate the induced EMF
Calculating the above expression:
\[ \epsilon = 0.2 \cdot 20 \cdot 4 \times 10^{-3} \]
\[ \epsilon = 1.6 \times 10^{-2} \, \text{V} \]

### Step 7: Convert to millivolts
To express the induced EMF in millivolts (mV):
\[ \epsilon = 1.6 \times 10^{-2} \, \text{V} = 16 \, \text{mV} \]

### Final Answer
The induced EMF in the rod is **16 mV**.

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