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A metal rod moves at a constant velocity...

A metal rod moves at a constant velocity in a direction perpendicular to its length. A constant, uniform magnetic field exists in space in a direction perpendicular to the rod as well as its velocity. Select the correct statements(s) from the following

A

The entire rod is at same electric potential

B

There is an electric field in the rod

C

The electric potential is highest at the center of the rod and decreases towards its ends

D

The electric potential is lowest at the center of the rod and increases towards its ends

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the scenario step by step. ### Step-by-Step Solution: 1. **Understanding the Setup**: - We have a metal rod of length \( l \) moving with a constant velocity \( v \). - The direction of the rod is perpendicular to its velocity. - A uniform magnetic field \( \vec{B} \) is also present, perpendicular to both the rod and its velocity. 2. **Identifying Charge Movement**: - Since the rod is made of metal, it contains free electrons that can move. - As the rod moves through the magnetic field, the electrons experience a magnetic force due to their motion in the field. 3. **Applying the Lorentz Force**: - The magnetic force \( \vec{F} \) on a charge \( q \) moving with velocity \( \vec{v} \) in a magnetic field \( \vec{B} \) is given by the equation: \[ \vec{F} = q (\vec{v} \times \vec{B}) \] - For electrons, the charge \( q \) is negative, which means the direction of the force will be opposite to that calculated for a positive charge. 4. **Determining the Direction of the Force**: - Using the right-hand rule, we can determine the direction of \( \vec{v} \times \vec{B} \): - Point your fingers in the direction of \( \vec{v} \) (the velocity of the rod). - Curl your fingers in the direction of \( \vec{B} \) (the magnetic field). - Your thumb will point in the direction of the force on a positive charge. - Since electrons are negatively charged, they will move in the opposite direction of the thumb. 5. **Charge Accumulation**: - As a result of the force, electrons will accumulate at one end of the rod, creating a negative charge at that end and a positive charge at the opposite end. - This leads to a separation of charge, resulting in an electric field being established within the rod. 6. **Electric Field Direction**: - The induced electric field \( \vec{E} \) will point from the positive charge (higher potential) to the negative charge (lower potential). 7. **Potential Difference**: - Since there is a charge separation, the two ends of the rod will have different electric potentials. - The end where electrons accumulate will have a lower potential, and the end where positive charges accumulate will have a higher potential. 8. **Conclusion**: - Based on the analysis, we can conclude: - There is an induced electric field in the rod. - The rod does not have the same electric potential throughout; there is a potential difference between the ends. ### Correct Statements: - The induced electric field exists in the rod (correct). - The entire rod is not at the same electric potential (incorrect). - The electric potential is highest at one end and lowest at the other end (correct).
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