Gauss' Theorem — Assertion and Reason Type

Assertion (A): No two electric lines of force can intersect each other.

Reason (R): Tangent at any point of electric line of force gives the direction of electric field.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: No two electric lines of force can intersect each other. If two lines of force were to intersect, then at the point of intersection two tangents could be drawn to the two lines of force. This would mean two directions of electric field intensity at the same point, which is not possible.

Reason (R) is also correct: The tangent drawn at any point of an electric line of force gives the direction of the electric field at that point. This is precisely why two intersecting lines would give two directions at one point.

The Reason therefore explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): The surface densities of two spherical conductors of different radii are equal. Then the electric field intensities near their surface are also equal.

Reason (R): Surface density is equal to charge per unit area and electric field near the surface is given by $\text E = \text k\dfrac{\text q}{\text r^2}$.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: For the two spherical conductors it is given that σ₁ = σ₂. Let r₁ and r₂ be their radii, so that

$\dfrac{\text q_1}{4\pi \text r_1^2} = \dfrac{\text q_2}{4\pi \text r_2^2} \\[1em] \dfrac{\text q_1}{\text q_2} = \dfrac{\text r_1^2}{\text r_2^2}$

Reason (R) is also correct: The surface density is the charge per unit area, and the electric field near the surface of a spherical conductor is $\text E = \text k\dfrac{\text q}{\text r^2}$. Taking the ratio of the intensities near the two surfaces,

$\dfrac{\text E_1}{\text E_2} = \dfrac{\text q_1}{4\pi\epsilon_0 \text r_1^2} \times \dfrac{4\pi\epsilon_0 \text r_2^2}{\text q_2} \\[1em] \dfrac{\text E_1}{\text E_2} = \dfrac{\text q_1}{\text q_2} \times \dfrac{\text r_2^2}{\text r_1^2}$

Substituting $\dfrac{\text q_1}{\text q_2} = \dfrac{\text r_1^2}{\text r_2^2}$,

$\dfrac{\text E_1}{\text E_2} = \dfrac{\text r_1^2}{\text r_2^2} \times \dfrac{\text r_2^2}{\text r_1^2} \\[1em] \dfrac{\text E_1}{\text E_2} = 1 \\[1em] \text E_1 = \text E_2$

The Reason therefore explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): Range of Coulomb force is infinite.

Reason (R): Coulomb force acts between two charged particles.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: The Coulomb force, or electric force, given by Coulomb's law is valid over nuclear distances (about 10^-15 m) as well as over very large distances. Hence the range of the Coulomb force is infinite.

Reason (R) is also correct: From Coulomb's law,

$\text F = \dfrac{\text k|\text q_1||\text q_2|}{\text r^2}$

the force does act between two charged particles. This generally holds only for charged objects whose sizes are much smaller than the distance between them.

However, the Reason merely states between what the Coulomb force acts; it says nothing about how far the force extends. The infinite range follows from the fact that the force falls off as $\dfrac{1}{\text r^2}$ and never becomes exactly zero at any finite distance. Hence the Reason does not explain the Assertion.

Therefore, both assertion and reason are true but reason is not the correct explanation of assertion.

Assertion (A): If a point charge be rotated in a circle around a charge, the work done will be zero.

Reason (R): Work done is equal to dot product of force and displacement. Angle between the two at any instant is 90°.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: When a point charge is rotated in a circle around another charge, the work done is zero.

Reason (R) is also correct: The work done is the dot product of force and displacement,

$\text W = \vec{\text F} \cdot \vec{\text S} = \text{FS}\cos\theta$

In rotating the charge in a circle, the electrostatic force is along the radius while the direction of motion is along the tangent, so the angle between the two at any instant is 90°. Therefore

$\text W = \text{FS}\cos 90^\circ \\[1em] \text W = 0$

The Reason therefore explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): If a conducting medium is placed between two charges, then electric force between them becomes zero.

Reason (R): Reduction in a force due to introduced material is inversely proportional to its dielectric constant.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: When a conducting medium is placed between two charges, the electric force between them becomes zero.

Reason (R) is also correct: The dielectric constant of any material is

$\text K = \dfrac{\text E_0}{\text E} = \dfrac{\text F_0}{\text F} \\[1em] \text F = \dfrac{\text F_0}{\text K}$

where F₀ is the force when no material is present between the charges and F is the force after the material is introduced. Thus the reduction in the force is inversely proportional to the dielectric constant.

Since the dielectric constant of a conducting medium is infinity,

$\text F = \dfrac{\text F_0}{\infty} \\[1em] \text F = 0$

The Reason therefore explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): In electrostatics, electric lines of force can never be closed loops.

Reason (R): The number of electric lines of force originating or terminating on a charge is proportional to the magnitude of charge.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: In electrostatics, the electric lines of force can never form closed loops. An electric line of force originates from a positive charge and terminates on a negative charge, so it cannot return to its starting point.

Reason (R) is also correct: The electric flux is a measure of the number of lines of force passing through a surface, and by Gauss' theorem the flux through a closed surface enclosing a charge q is $\dfrac{\text q}{\epsilon_0}$. Hence the number of lines of force originating or terminating on a charge is proportional to the magnitude of the charge.

However, the Reason concerns only how many lines of force a charge has; it says nothing about whether a line of force can close on itself. Hence the Reason does not explain the Assertion.

Therefore, both assertion and reason are true but reason is not the correct explanation of assertion.

Note: The printed key gives option 3, treating the Reason as false. The Reason is a true statement but does not explain the Assertion, so the correct option is 2.

Assertion (A): Charge is quantized.

Reason (R): Charge which is less than 1 C is not possible.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: Charge is quantized. The charge q on a body is always given by

$\text q = \text{ne}$

where n is any positive or negative integer and e is the elementary charge.

Reason (R) is incorrect: The charge on an electron is e = 1.6 × 10^-19 C, which is very much less than 1 C. Hence it is not true that a charge less than 1 C is not possible. Quantization means that the charge must be an integral multiple of e, not that it must exceed 1 C.

Therefore, assertion is true but reason is false.

Assertion (A): In a cavity within a conductor, the electric field is zero.

Reason (R): Charges in a conductor reside only at its surface.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: In a cavity within a conductor, the electric field is zero.

Reason (R) is also correct: The charge given to a conductor resides only at its surface.

Since the charges reside only on the outer surface of the conductor, no charge resides on the wall of the empty cavity. An electric line of force must begin on a positive charge and end on a negative charge, and in electrostatics it cannot form a closed loop; with no charge inside the cavity or on its wall, no line of force can exist within the cavity. Moreover, the wall of the cavity is a part of the conductor, so all its points are at the same potential, whereas a line of force always runs from a higher to a lower potential. Hence the electric field in the cavity is zero.

The Reason therefore explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Note: The printed explanation infers E = 0 from zero flux through a Gaussian surface, although the chapter states that zero flux does not imply E = 0; the keyed option 1 is nevertheless correct.

Assertion (A): An electrostatic field line never form closed loop.

Reason (R): Electrostatic field is a conservative field.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: An electrostatic field line never forms a closed loop. The field originates from a positive charge and terminates on a negative charge, and since isolated positive and negative charges do exist, the field lines do not close on themselves.

Reason (R) is also correct: The electrostatic field is a conservative field, which means that the work done by an electrostatic field in moving a charge from one point to another depends only on the initial and the final positions and is independent of the path taken.

Since the work done depends only on the initial and the final positions, the work done by the field in taking a charge round any closed path is zero. If a field line formed a closed loop, a positive charge taken round that loop along the line would have the force on it in the direction of its motion at every point, so the work done would be positive and not zero. Hence the conservative nature of the field rules out closed field lines, and the Reason explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Note: The printed key gives option 2. Since a closed field line would make the work done round it non-zero, contrary to the conservative nature of the field, the Reason explains the Assertion and the correct option is 1.

Assertion (A): If a conductor is given charge then no excess inner charge appears.

Reason (R): Electric field inside conductor is zero.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: When a conductor is given a charge, no excess charge appears inside it; the excess charge can reside only on its surface.

Reason (R) is also correct: The electric field inside a conductor is zero.

Since the electric field inside the conductor is zero, the flux through any Gaussian surface drawn wholly within the material of the conductor is zero (E = 0 implies φ = 0). By Gauss' theorem the charge enclosed by every such surface is then zero, so no excess charge can appear inside the conductor and it resides only on the surface. The Reason therefore explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Note: The printed key gives option 2. Since the zero field inside the conductor is the very premise from which Gauss' theorem shows that no inner charge appears, the Reason explains the Assertion and the correct option is 1.

Assertion (A): A charged particle free to move in an electric field always moves along an electric field line.

Reason (R): The electric field lines diverge from a negative charge and converge at a positive charge.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: Only a charged particle which is initially at rest moves along the electric field line. If the charged particle is already in motion and its initial velocity is at an angle with the electric field line, the particle may follow a parabolic path, just as a projectile does.

Reason (R) is also incorrect: The electric field lines diverge from a positive charge and converge at a negative charge, and not the other way round as stated.

Therefore, both assertion and reason are false.

Assertion (A): If an electron and proton possessing same kinetic energy enter an electric field in a perpendicular direction, the path of the electron is more curved than that of the proton.

Reason (R): Electron forms a larger curve due to its small mass.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: When a charged particle enters an electric field in a perpendicular direction, it describes a path of radius of curvature r given by

$\text{qE} = \dfrac{\text{mv}^2}{\text r} \\[1em] \text r = \dfrac{\text{mv}^2}{\text{qE}} \\[1em] \text r = \dfrac{2(\text{K.E.})}{\text{qE}} \qquad \left(\because \text{K.E.} = \dfrac{1}{2}\text{mv}^2\right)$

so that $\text r \propto \dfrac{1}{\text q}$. The electron and the proton carry charges of equal magnitude and are given the same kinetic energy, so r is the same for both of them. Hence the path of the electron is not more curved than that of the proton.

Reason (R) is also incorrect: As the above relation shows, r depends upon the kinetic energy, which is the same for both the particles, and not upon their masses. So the smaller mass of the electron does not make its curve larger.

Therefore, both assertion and reason are false.

Assertion (A): Acceleration of charged particle in non-uniform electric field does not depend on velocity of charged particle.

Reason (R): Charge is an invariant quantity. That is, amount of charge on particle does not depend on frame of reference.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: The acceleration of a charged particle in an electric field is

$\vec{\text a} = \dfrac{\text q\vec{\text E}}{\text m}$

In a non-uniform field $\vec{\text E} \space$varies from point to point, and so does $\vec{\text a}$. But the acceleration depends only on the field at the location of the particle and on its charge and mass; it does not depend on the velocity of the charged particle.

Reason (R) is also correct: Charge is an invariant quantity, that is, the amount of charge on a particle does not depend on the frame of reference.

The invariance of charge is a separate property and is not the reason why the acceleration is independent of velocity; that follows from the relation $\vec{\text a} = \dfrac{\text q\vec{\text E}}{\text m}$ itself. Hence the Reason does not explain the Assertion.

Therefore, both assertion and reason are true but reason is not the correct explanation of assertion.

Assertion (A): A negative charge in an electric field moves along the direction of the electric field.

Reason (R): On a negative charge a force acts in the direction of the electric field.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: A negative charge placed in an electric field moves along the direction opposite to that of the electric field, and not along the direction of the field.

Reason (R) is also incorrect: The force on a charge q in a field $\vec{\text E} \space$is $\vec{\text F} = \text q\vec{\text E}$. For a negative charge this becomes

$\vec{\text F} = -\text q\vec{\text E}$

so the force acts in the direction opposite to the electric field, and not in the direction of the electric field.

Therefore, both assertion and reason are false.

Assertion (A): Gauss' law shows diversion when inverse square law is not obeyed.

Reason (R): Gauss' law is a consequence of conservation of charges.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: Gauss' theorem is valid for all vector fields obeying the inverse square law. It is based on the inverse square dependence of the electric field on distance, so if the inverse square law were not obeyed, Gauss' law would show a diversion, that is, it would no longer hold.

Reason (R) is also correct: Charge is conserved, and Gauss' theorem is consistent with the conservation of charge.

However, the validity of Gauss' theorem follows from the inverse square dependence of the field on distance, and not from the conservation of charge. Hence the Reason does not explain the Assertion.

Therefore, both assertion and reason are true but reason is not the correct explanation of assertion.

Assertion (A): Gauss' law can't be used to calculate electric field near an electric dipole.

Reason (R): Electric dipole don't have symmetrical charge distribution.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Gauss' law cannot conveniently be used to calculate the electric field near an electric dipole.

Reason (R) is also correct: Gauss' law is applicable for any closed surface, but it is most useful where the charge distribution has spherical or cylindrical symmetry, or is distributed uniformly over a plane. An electric dipole is a system of two equal and opposite point charges separated by a very small and finite distance, and such a charge distribution is not symmetrical.

Since the field of a dipole is neither constant in magnitude nor everywhere normal to any convenient closed surface, E cannot be taken out of the integral $\oint \vec{\text E} \cdot \text d\vec{\text A}$. The Reason therefore explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): A point charge is lying at the centre of a cube of each side. The electric flux emanating from each surface of the cube is $\dfrac{1}{6}$ th of total flux.

Reason (R): Gauss' law is very helpful in calculating electric field in those cases where electric field is symmetrical around the source producting it.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: By Gauss' theorem, the electric flux through the cube is $\phi = \dfrac{\text q}{\epsilon_0}$. A cube has six faces of equal area, and with the point charge at the centre they are all symmetrically situated with respect to it. Hence the flux through each face is

$\dfrac{1}{6}\phi = \dfrac{1}{6}\left(\dfrac{\text q}{\epsilon_0}\right)$

Reason (R) is also correct: Gauss' law is very helpful in calculating the electric field in those cases where the electric field is symmetrical around the source producing it.

However, the Reason concerns the calculation of the electric field from Gauss' law, whereas the Assertion follows from the symmetry of the six faces of the cube about the central charge. Hence the Reason does not explain the Assertion.

Therefore, both assertion and reason are true but reason is not the correct explanation of assertion.

Assertion (A): The electric flux emanating out and entering a closed surface are 8 × 10³ and 2 × 10³ V-m respectively. The charge enclosed by the surface is 0.053 μC.

Reason (R): Gauss' theorem in electrostatics may be applied to verify.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The flux emerging out is 8 × 10³ V-m and that entering is 2 × 10³ V-m. Taking the outgoing flux as positive and the incoming flux as negative, the net flux through the closed surface is

$\phi = 8 \times 10^3 - 2 \times 10^3 \\[1em] \phi = 6 \times 10^3\ \text{V-m}$

Reason (R) is also correct: Gauss' theorem in electrostatics may indeed be applied to verify the result. According to it,

$\phi = \dfrac{\text q}{\epsilon_0} \\[1em] \text q = \epsilon_0 \phi$

Substituting the values,

$\text q = (8.85 \times 10^{-12})(6 \times 10^3) \\[1em] = 53.10 \times 10^{-9}\ \text C \\[1em] \text q = 0.053\ \mu\text C$

which is the value stated in the Assertion. The Reason therefore explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

Assertion (A): If an electric dipole of dipole moment 30 × 10^-5 C-m is enclosed by a closed surface, the net flux coming out of the surface will be zero.

Reason (R): Electric dipole consists of two equal and opposite charges.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: When an electric dipole is enclosed by a closed surface, the net flux coming out of the surface is zero, whatever the value of the dipole moment.

Reason (R) is also correct: An electric dipole consists of two equal and opposite charges separated by a small distance.

Since the two charges are equal and opposite, the net charge enclosed by the surface is zero. By Gauss' theorem the net flux is $\dfrac{1}{\epsilon_0}$ times the charge enclosed,

$\Phi_\text E = \dfrac{0}{\epsilon_0} \\[1em] \Phi_\text E = 0$

The Reason therefore explains the Assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.