Electric Charges and Fields

A charge Q is placed on an isolated conducting sphere of radius R, storing electrostatic energy U₁ = kQ²/(2R). Instead, if the same charge Q is placed on the inner sphere of a spherical capacitor formed by adding a concentric, grounded, outer conducting shell of radius 2R around it, the new stored energy U₂ is:

1. U₁/2
2. U₁/4
3. 2U₁
4. U₁, unchanged

A dielectric slab of thickness t (t < d) and dielectric constant κ is inserted between the plates of a parallel plate capacitor of separation d, using the general formula C = ε₀A / [d − t(1 − 1/κ)]. In the limit κ → ∞, the capacitance:

1. approaches infinity, since κ → ∞ implies unbounded polarization
2. approaches ε₀A/(d − t), exactly identical to inserting a conducting slab of the same thickness t
3. approaches zero, since an infinitely polarizable medium completely blocks the field
4. remains indeterminate as κ → ∞

A short electric dipole of moment p is fixed at the origin, oriented along the x-axis. A point charge +q is placed on the y-axis at (0, r) — on the equatorial line of the dipole. The electrostatic potential ENERGY of interaction between the dipole and the charge is:

1. kpq/r²
2. zero
3. −kpq/r²
4. kpq/(2r²)

A parallel plate capacitor, isolated (disconnected from any battery) after being charged to potential V₀, has its plate separation doubled (d → 2d) AND its plate area simultaneously doubled (A → 2A) by an overlapping slide mechanism. The new potential difference across the capacitor is:

1. V₀/2
2. 2V₀
3. V₀, exactly unchanged
4. 4V₀

For a uniformly charged solid sphere of total charge Q and radius R, the total electrostatic self-energy is U_total = 3kQ²/(5R). The fraction of this total self-energy that resides OUTSIDE the sphere (in the region r > R) is:

1. 1/6
2. 5/6
3. 1/2
4. 3/5

Two concentric conducting spherical shells have inner radius a with charge +Q, and outer radius b with charge +2Q (not −Q as in the standard spherical capacitor). The potential difference V_a − V_b is:

1. kQ(1/a − 1/b), exactly the same as if the outer shell carried −Q instead — completely unaffected by the outer shell's own charge
2. kQ(1/a − 1/b) − k(2Q)/b, since the outer shell's charge must be subtracted
3. k(3Q)(1/a − 1/b), since total enclosed charge at radius a is effectively 3Q
4. cannot be determined without knowing whether the shells are grounded

A dielectric slab of constant κ, with breadth equal to the plate width and thickness equal to the full plate separation d, is partially inserted to a depth x into a parallel plate capacitor (plate length L, width b) connected to a battery maintaining constant voltage V. As x increases from 0 to L, the force pulling the slab further into the capacitor:

1. increases proportionally with x
2. decreases as x increases, approaching zero near full insertion
3. remains constant, completely independent of x, as long as V is held fixed
4. is zero for all x, since dielectrics are not electrically attracted into capacitors

Five identical capacitors, each of capacitance C, are connected in a bridge network with the same topology as a Wheatstone bridge: four capacitors form the four arms, and the fifth connects the two midpoints (the "bridge" capacitor). The charge on the bridge (fifth) capacitor, once the network reaches steady state, is:

1. equal to the charge on any arm capacitor
2. zero, since equal arm capacitances automatically satisfy the balance condition, exactly analogous to a balanced Wheatstone bridge in resistors
3. cannot be determined without knowing the applied voltage
4. infinite, since capacitors block steady current, making the bridge charge indeterminate

A parallel plate capacitor remains connected to a battery of constant EMF V₀ throughout. A dielectric slab of constant κ is then inserted completely, filling the gap. Which one of the following quantities remains UNCHANGED?

1. Capacitance C
2. Charge Q on the plates
3. Electric field E between the plates
4. Energy U stored in the capacitor

Capacitor C₁ = 2 μF is charged to 10 V, and capacitor C₂ = 4 μF is charged to 5 V (both have 20 μC of charge, but note C₁'s positive plate holds +20 μC). They are now connected such that the positive plate of C₁ is joined to the NEGATIVE plate of C₂ (opposite polarity connection). The common potential and heat dissipated are:

1. V = 2.5 V; heat = 0 J
2. V = 0 V; heat = 150 μJ, i.e., the entire initial stored energy is dissipated
3. V = 2.5 V; heat = 75 μJ
4. V = 5 V; heat = 50 μJ

A parallel plate capacitor of plate area A and separation d is filled with two dielectric slabs of constants κ₁ and κ₂, each of thickness d, placed side by side so that each occupies half the plate area (A/2), with the boundary between them running parallel to the electric field lines. This arrangement behaves as:

1. two capacitors in series: C = ε₀A / [d(1/κ₁ + 1/κ₂)]
2. two capacitors in parallel, each of area A/2: C = (ε₀A/2d)(κ₁ + κ₂)
3. neither series nor parallel; the field distribution requires solving Laplace's equation
4. a single capacitor with an averaged dielectric constant (κ₁+κ₂)/2

A physical (real) electric dipole consists of point charges +q and −q separated by a finite distance 2a (not necessarily small). The surface on which the electrostatic potential is exactly zero is:

1. only approximately the perpendicular bisector plane, becoming exact only in the limit r → ∞
2. exactly the perpendicular bisector plane of the line joining the two charges, at every distance, with no approximation involved
3. a curved, non-planar surface for finite r, flattening into a plane only when r ≫ a
4. nonexistent for a physical dipole; V = 0 occurs only at infinity

Point charges +2q and −q are fixed at x = 0 and x = d respectively on the x-axis. There exists a point (other than at infinity) on the x-axis, between the two charges, where the electric potential V = 0. At this point, the electric field is:

1. zero, since V = 0 automatically implies E = 0 there
2. zero, but only coincidentally at this particular point
3. nonzero, since E = −dV/dx depends on the local slope of V, not on the value of V itself
4. infinite, since the point lies on the segment joining the two charges

A solid conductor of irregular shape carries charge and has a uniform surface potential V₀ everywhere on its surface. The potential at the geometric centre of the solid interior is:

1. zero, since the electric field is zero throughout the interior
2. V₀, the same as the surface, since E = 0 inside implies no potential change occurs while moving through the interior
3. less than V₀ but nonzero, depending on the exact shape
4. cannot be determined without knowing the shape

Three equal point charges +q are fixed at the vertices of an equilateral triangle of side a. One charge is now slowly moved to infinity by an external agent, while the other two remain fixed. The work done BY THE EXTERNAL AGENT in this process is:

1. +2kq²/a
2. −2kq²/a
3. +3kq²/a
4. zero, since the charge simply moves away under its own repulsion

In Coulomb's law, F = kq1q2/r^2, if both charges q1 and q2 are doubled AND the distance r between them is also doubled, while simultaneously the entire system is submerged in a dielectric medium of relative permittivity εr = 4 (replacing vacuum), the new force compared to the original force F (in vacuum) is:

1. F, unchanged
2. F/4
3. 4F
4. F/16

A charge +Q is uniformly distributed along a quarter-circular arc of radius R, lying in the first quadrant (from the positive x-axis to the positive y-axis). The direction of the net electric field at the centre O of the circle (origin) is:

1. along the positive x-axis
2. along the positive y-axis
3. along the line y = x, pointing into the third quadrant (i.e., away from the arc)
4. along the line y = x, pointing into the first quadrant (i.e., toward the arc)

Two thin concentric spherical shells of radii R and 2R carry charges +Q and -Q respectively (Q on inner shell, -Q on outer shell). The electric field at a distance r, where R 2R, respectively, are:

1. kQ/r² (inward) for R 2R
2. kQ/r² (outward) for R 2R
3. zero for R 2R
4. kQ/r² (outward) for R 2R

A charged particle of charge q and mass m enters a region of uniform electric field E (directed vertically) with an initial horizontal velocity u, similar to horizontal projectile motion under gravity. If the field is suddenly reversed in direction after the particle has traveled exactly half of its total vertical deflection (at the original field's settings, for a fixed horizontal distance L), the particle's net vertical displacement at horizontal distance L compared to the case with no reversal is:

1. the same, since the total impulse over time is unchanged
2. exactly half, since the field acts oppositely for the second half of the time
3. exactly half of what it would have been, but must be computed via time intervals, not distance -- giving a value different from simply 'half'
4. zero, since the reversal exactly cancels the first half's effect

An infinite line charge with linear charge density λ passes through the centre of a spherical Gaussian surface of radius R, such that the line passes through two diametrically opposite points on the sphere. The total electric flux through the sphere is:

1. λ(2R)/ε₀, since flux depends on enclosed charge and the enclosed length is the diameter 2R
2. infinite, since the line charge extends infinitely
3. λR/ε₀
4. zero, since a line has no volume and hence no enclosed 'point' charge

A charge q is placed at the centre of a cube. Now, instead, the same charge q is placed at the centroid of a regular tetrahedron (4 triangular faces). The flux through one face of the tetrahedron is:

1. q/6ε₀, same as the cube case for one face
2. q/4ε₀
3. q/3ε₀
4. cannot be determined without knowing the tetrahedron's edge length

Two point charges +q and +q are fixed at (-a, 0) and (+a, 0). A third charge -q is placed at the origin. If the third charge is displaced slightly along the y-axis (perpendicular to the line joining the fixed charges) and released, its motion is:

1. simple harmonic, since the origin is a stable equilibrium in all directions
2. simple harmonic only for this y-axis displacement, though the origin is unstable for displacement along the x-axis
3. the charge simply moves away and never returns, since equilibrium is unstable everywhere
4. the charge remains stationary at any small displacement, since net force is always zero along the y-axis

A solid non-conducting sphere of radius R carries a uniform volume charge density ρ. A spherical cavity of radius R/2 is carved out of it such that the cavity's surface passes through the centre of the original sphere (i.e., the cavity is tangent internally, touching the centre). The electric field at the centre O of the original sphere is:

1. zero, by symmetry
2. ρR/(6ε₀), directed from the centre of the cavity toward O
3. ρR/(3ε₀), directed from O toward the centre of the cavity
4. ρR/(6ε₀), directed from O toward the centre of the cavity

Consider an electric dipole placed in a uniform electric field E, initially aligned along E (θ = 0). The work done by an external agent to rotate the dipole to θ = 180 degrees is:

1. zero, since the dipole returns to being aligned (anti-aligned) with the field
2. 2pE
3. pE
4. -2pE, since the process is spontaneous and releases energy

A charge Q is uniformly distributed over a thin, non-conducting ring of radius R. A small negative test charge -q (mass m) is placed at the centre of the ring and displaced slightly along the axis by a small distance x (x << R), then released. The subsequent motion is:

1. uniformly accelerated motion away from the centre
2. simple harmonic motion, since restoring force is proportional to x for x << R
3. oscillatory but not simple harmonic, since force depends on (R²+x²)^(3/2)
4. the charge remains in equilibrium and does not move, since the centre is a stable equilibrium at all displacements

Two infinite plane sheets of charge, with uniform surface densities +σ and +σ (same sign, same magnitude), are placed parallel to each other. The electric field in the region between the two sheets is:

1. σ/ε₀, directed from one sheet to the other
2. σ/(2ε₀), directed toward either sheet symmetrically... which is impossible, hence zero
3. zero, since fields due to both sheets point in opposite directions between them and cancel
4. 2σ/ε₀, since fields add up between the sheets

A hollow spherical conductor of inner radius a and outer radius b carries a net charge Q. A point charge q is placed at the centre of the cavity. The surface charge density on the outer surface of the conductor is:

1. Q/(4πb²)
2. (Q+q)/(4πb²)
3. (Q-q)/(4πb²)
4. q/(4πb²), independent of Q

An electric dipole of moment p is placed in a non-uniform electric field E = kx (along x-axis), with the dipole also aligned along the x-axis, centered at x = x0. The correct statement is:

1. The dipole experiences only a torque, no net force
2. The dipole experiences a net force but zero torque, since it is aligned with the field
3. The dipole experiences both a net force and zero torque; torque is zero because sinθ = 0
4. The dipole experiences neither force nor torque since E is linear in x

A point charge +q is placed at the centre of a cube of side a. If the charge is now shifted to one corner of the cube (still inside), the electric flux through any one face of the cube:

1. remains q/6ε₀ for every face
2. becomes zero for three faces meeting at that corner
3. becomes q/24ε₀ for the three faces sharing that corner and varies for the rest, but total remains q/ε₀
4. becomes undefined since charge lies on the boundary

Two identical small conducting spheres A and B carry charges +12 uC and -4 uC respectively and are separated by a distance r, experiencing a force of magnitude F (attractive). The spheres are first brought into contact and then separated back to the same distance r. Each sphere is then doubled in radius (charge redistributes uniformly on each sphere; total charge on each is conserved during resizing). What is the new force, compared to F?

1. F/12
2. F/4
3. F/3
4. 3F/4