A 20 cm long metallic rod is rotated with 210 rpm about an axis normal to the rod passing through its one end. The order end of the rod is in contact with a circular metallic ring. A constant and uniform magnetic field 0.2T parallel to the axis exists everywhere. The emf developed between the centre and the ring is ____ mV.
Correct Answer: 88
Solution and Explanation
We are given the length of a rod and its angular speed. We need to find the induced emf across the ends of the rod in a magnetic field.
Solution
1. Given Values:
Length of the rod, \( \ell = 20 \, \text{cm} = 0.2 \, \text{m} \)
Angular speed, \( \omega = 210 \, \text{rpm} \)
Magnetic field, \( B = 0.2 \, \text{T} \) (assumed from the calculation)
2. Convert rpm to Radians per Second:
\( \omega = 210 \times \frac{2\pi}{60} = 210 \times \frac{\pi}{30} = 7\pi \, \text{rad/s} \)
Using the approximate value of π as 22/7:
\( \omega = 7 \times \frac{22}{7} = 22 \, \text{rad/s} \)
3. Use the Formula for Induced EMF:
The induced emf is given by:
\( \text{emf} = \frac{1}{2} B \omega \ell^2 \)
4. Substitute the Known Values:
\( \text{emf} = \frac{1}{2} \times 0.2 \times 22 \times (0.2)^2 \)
\( \text{emf} = 0.1 \times 22 \times 0.04 \)
\( \text{emf} = 2.2 \times 0.04 \)
\( \text{emf} = 0.088 \, \text{V} \)
\( \text{emf} = 88 \, \text{mV} \)
Final Answer
Thus, the induced emf is 88 mV.
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