An \( \alpha \)-particle of mass \( 6.4 \times 10^{-27}\text{ kg} \) and charge \( 3.2 \times 10^{-19}\text{ C} \) is situated in a uniform electric field of \( 1.6 \times 10^5\text{ Vm}^{-1} \). The velocity of the particle at the end of \( 2 \times 10^{-2}\text{ m} \) path when it starts from rest is:
1. \( 2\sqrt{3} \times 10^5\text{ ms}^{-1} \)
2. \( 8 \times 10^5\text{ ms}^{-1} \)
3. \( 16 \times 10^5\text{ ms}^{-1} \)
4. \( 4\sqrt{2} \times 10^5\text{ ms}^{-1} \)
View Answer
Using the work-energy theorem, \( qEd = \frac{1}{2}mv^2 \). Rearranging gives \( v = \sqrt{\frac{2qEd}{m}} = \sqrt{\frac{2 \times 3.2 \times 10^{-19} \times 1.6 \times 10^5 \times 2 \times 10^{-2}}{6.4 \times 10^{-27}}} = 4\sqrt{2} \times 10^5\text{ m/s} \).
Assertion: Sharper is the curvature of spot on a charged body lesser will be the surface charge density at that point.
Reason: Electric field is non-zero inside a charged conductor.
1. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
2. Both Assertion and Reason are true but Reason is not correct explanation of Assertion.
3. Assertion is true but Reason is false.
4. Assertion and Reason are false.
View Answer
Surface charge density is directly proportional to curvature \(\sigma \propto 1/R\). Inside a conductor, the electric field is zero. Hence, both statements are false.
A regular polygon has \(n\) sides each of length \(l\). Each corner of the polygon is at a distance \(r\) from the centre. Identical charges each equal to \(q\) are placed at all the corners except one. The magnitude of electric field at the centre of the polygon is
1. \(\frac{kq(n-1)}{r^2}\)
2. \(\frac{2kq}{r^2}\)
3. \(\frac{kqn}{r^2}\)
4. \(\frac{kq}{r^2}\)
View Answer
A completely symmetric distribution of \(n\) charges has a net field of zero at the center. Removing one charge is equivalent to superimposing a charge of \(-q\) at the empty corner on an otherwise full polygon, which produces a field of magnitude \(\frac{kq}{r^2}\).
Identify the incorrect statement among the following
1. Electrostatic field at the surface of a charged conductor is proportional to the surface charge density
2. There is no net charge at any point inside conductor when a charge is placed outside it
3. Inside charged or neutral conductor, electrostatic field is zero
4. The electrostatic field at surface of charged conductor must be tangential to surface at any point when placed in external electric field
View Answer
Under electrostatic conditions, the electric field at the surface of a charged conductor must be perpendicular (normal) to the surface at every point. It cannot be tangential, otherwise charges would flow along the surface.
Assertion (A): A metal sphere of radius \(1\text{ cm}\) cannot hold a charge of \(1\text{ coulomb}\) in air.
Reason (R): The dielectric strength of air (minimum field required for ionisation of a medium) is \(3\text{ MV/m}\).
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
View Answer
Assertion (A) is true: \(1\text{ C}\) is a huge charge for a \(1\text{ cm}\) sphere.
Reason (R) is true: Air's dielectric strength is \(3 \times 10^6\text{ V/m}\). The electric field at the surface \(E = \frac{Q}{4\pi \epsilon_0 R^2}\text{ }\approx 9 \times 10^{13}\text{ V/m}\).
This field greatly exceeds air's dielectric strength, causing electrical breakdown. Thus, (R) explains (A).
Assertion (A): In any electrostatic field, a charge cannot be in stable equilibrium.
Reason (R): An electrostatic field is a conservative force field.
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
View Answer
According to Earnshaw's Theorem, a charge cannot be in stable equilibrium in an electrostatic field, thus Assertion (A) is true. An electrostatic field is a conservative force field, so Reason (R) is true. However, the conservative nature of the field is not the direct explanation for Earnshaw's theorem, which is derived from Gauss's law and the Laplace equation.
Assertion (A): If a proton and an electron are placed in the same uniform electric field one by one, they experience different accelerations (The only force acting on proton and electron is that exerted by uniform electric field).
Reason (R): Electric force on a test charge is independent of its mass.
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
View Answer
Assertion (A) is true because \(a = F/m\). In a uniform electric field \(E\), the magnitude of force on both proton and electron is \(F = eE\). Since their masses are different (\(m_e \neq m_p\)), their accelerations \(a\) will be different. Reason (R) is true as the electric force \(F = qE\) depends on charge \(q\) and electric field \(E\), not mass \(m\). Reason (R) correctly explains Assertion (A).
Assertion (A): When a negative charge \(-q\) is released at a distance \(R\) from the centre and along the axis of a uniformly and positively charged fixed ring of radius \(R\), the negative charge does oscillation but not SHM.
Reason (R): The force on negative charge is always towards the centre of the ring but it is not proportional to the displacement from the centre of the ring.
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
View Answer
Assertion (A) is true. The electric force on charge \(-q\) on the axis of a positively charged ring is a restoring force towards the center, causing oscillation. The force is \(F = \frac{kQqx}{(R^2+x^2)^{3/2}}\). This is not linearly proportional to \(x\) (displacement) unless \(x \ll R\), so it's not SHM. Reason (R) is true. The force is attractive (towards center) and indeed not proportional to \(x\). Reason (R) correctly explains Assertion (A).
Assertion (A): There is an isolated system of two charged conducting spheres A and B. The resultant electric field at point P is the sum of electric field at P due to charged sphere A only (that is, assuming sphere B and all its effects to be absent) and the electric field at P only due to sphere B (that is, assuming sphere A and all its effects to be absent).
Reason (R): Superposition theorem for electric field due to point charges states that resultant electric field at a point due to point charges is the sum of electric field at that point due to individual charges.
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
View Answer
The electric field \(\vec{E}\) obeys the superposition principle. Assertion (A) accurately describes this principle for fields from multiple charge distributions. Reason (R) correctly states the superposition theorem. Hence, R correctly explains A.