Assertion (A): Electrostatic field inside a conducting shell is always zero.
Reason (R): The electrostatic potential is always same from center to surface of a conducting shell.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: Properties of conductors in electrostatic equilibrium.
Formula: \( \vec{E} = -\nabla V \).
Solution: In a conductor, free charges redistribute to make \( \vec{E} = 0 \) inside (A is true). If \( \vec{E} = 0 \) inside, then \( V \) must be constant (R is true) and thus (R) correctly explains (A).
The electrostatic potential on the surface of a charged solid conducting sphere is 100 volts. Two statements are made in this regard :
Assertion (A): At any point inside the sphere, electrostatic potential is 100 volt.
Reason (R): At any point inside the sphere, electric field is zero.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: Properties of charged conductors.
Principle: Inside a conductor, \( \vec{E} = 0 \) and \( V = \text{constant} \).
Solution: For a solid conducting sphere, potential is uniform throughout its volume and equals surface potential (A is true). This is because the electric field inside a conductor is zero (R is true), implying constant potential. Thus, (R) correctly explains (A).
Assertion (A): If electric field in x-y plane is given by \( \vec{E} = y\hat{i} + x\hat{j} \) then equipotential curve is given by \( xy = \text{ constant} \).
Reason (R): Electric field may not be perpendicular to equipotential surface/curve/line.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: Relation between electric field and equipotential surfaces.
Formula: \( \vec{E} = -\nabla V \).
Solution: From \( \vec{E} = -\nabla V \) for \( \vec{E} = y\hat{i} + x\hat{j} \), potential is \( V = -xy + C \), so equipotential curves are \( xy = \text{ constant} \) (A is true). Electric field lines are always perpendicular to equipotential surfaces (R is false).
Assertion (A): If electric field in x-y plane is given by \( \vec{E} = y \hat{i} + x \hat{j} \) then equipotential curve is given by \( xy = \text{constant} \).
Reason (R): Electric field may not be perpendicular to equipotential surface/curve/line.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For \( \vec{E} = y \hat{i} + x \hat{j} \), we have \( dV = -\vec{E} \cdot d\vec{l} = -(y dx + x dy) = -d(xy) \). Integrating, \( V = -xy + C \). Thus, equipotential lines are \( xy = \text{constant} \). Electric field lines are always perpendicular to equipotential surfaces. Thus (A) is true and (R) is false.
Assertion (A): When an isolated charged body is connected to earth, all its charge flows to earth and it becomes electrically neutral.
Reason (R): Electric potential of earth is non zero, so the body connected to earth should also attain zero potential.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Earth is considered to have a zero potential. When a charged body is connected to Earth, charges flow until the body also attains zero potential, making it electrically neutral. Thus (A) is true and (R) is false.
Assertion (A): Potential difference between two points in space is zero if electric field at all points in space is zero.
Reason (R): Electric field \( \vec{E} \) at a point \( P \) is zero if potential at that point is zero.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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If \( \vec{E} = \vec{0} \) everywhere, then \( \vec{E} = -\nabla V = \vec{0} \), implying \( V \) is constant, so \( \Delta V = 0 \). However, \( V = 0 \) at a point does not imply \( \vec{E} = \vec{0} \) (e.g., at the center of an electric dipole). Thus (A) is true and (R) is false.
Assertion (A): Electrostatic field inside a conducting shell is always zero.
Reason (R): The electrostatic potential is always same from center to surface of a conducting shell.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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In electrostatic equilibrium, the electric field inside a conductor is zero \( (\vec{E} = \vec{0}) \). Since \( \vec{E} = -\nabla V \), if \( \vec{E} = \vec{0} \), the potential \( V \) must be constant throughout the conductor, from center to surface. Thus (A) and (R) are true, and (R) explains (A).
The electrostatic potential on the surface of a charged solid conducting sphere is \( 100 \text{ volts} \). Two statements are made in this regard :
Assertion (A): At any point inside the sphere, electrostatic potential is \( 100 \text{ volt} \).
Reason (R): At any point inside the sphere, electric field is zero.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For a charged solid conducting sphere, the electric field inside is zero \( (\vec{E} = \vec{0}) \). Consequently, the potential \( V \) inside is constant and equal to the potential on its surface. Therefore, both (A) and (R) are true, and (R) correctly explains (A).
Assertion (A): Electric potential of earth is taken as zero.
Reason (R): Electric field strength on the surface of earth is zero.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Assertion (A): For practical purposes in circuit analysis and grounding, the Earth's electric potential is taken as zero because it is a large conductor and can absorb or supply charge without significant potential change. So (A) is true.
Reason (R): The electric field strength on the Earth's surface is generally not zero; it varies due to atmospheric conditions, presence of charges, etc. So (R) is false.nTherefore, (A) is true and (R) is false.