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Question 1Mathematical Physics

For the position vector r = x i + y j + z k in three-dimensional Cartesian coordinates, the divergence ∇·r is

  1. 1
  2. 0
  3. 3 Correct answer.
  4. r

Explanation

In Cartesian coordinates ∇·r = ∂x/∂x + ∂y/∂y + ∂z/∂z = 1 + 1 + 1 = 3. The result is a constant scalar, independent of the point, which reflects the uniform outward flux of the radial field.

Question 2Mathematical Physics

The residue of f(z) = 1/z at the point z = 0 is

  1. 1 Correct answer.
  2. 0
  3. 2πi
  4. ∞

Explanation

The Laurent expansion of 1/z about z = 0 is already the single term z^(-1), so the coefficient of z^(-1) is 1. By definition that coefficient is the residue at the simple pole.

Question 3Mathematical Physics

If M is a real orthogonal matrix, then det M must equal

  1. 1
  2. 0
  3. −1
  4. ±1 Correct answer.

Explanation

Orthogonality means MᵀM = I, so (det M)² = det(MᵀM) = det I = 1, giving det M = ±1. Rotations realise +1 and reflections realise -1, so both signs occur in general.

Question 4Mathematical Physics

The value of the integral ∫₀^{2π} dθ/(2 + cos θ) is

  1. 2π/√3 Correct answer.
  2. 2π√3
  3. π/√3
  4. 2π/3

Explanation

For a > |b| the standard result is ∫₀^{2π} dθ/(a + b cos θ) = 2π/√(a² − b²). Taking a = 2 and b = 1 gives 2π/√3, as can also be checked by the substitution z = e^{iθ} and the residue at z = -2 + √3.

Question 5Mathematical Physics

A particular integral of the equation y″ − y = e^{2x} is

  1. (1/3)e^{2x} Correct answer.
  2. e^{2x}
  3. (1/2)e^{2x}
  4. x e^{2x}

Explanation

Substituting y_p = A e^{2x} gives (4A − A)e^{2x} = 3A e^{2x}, which must equal e^{2x}, hence A = 1/3. The forcing e^{2x} is not a solution of the homogeneous equation y″ − y = 0, so no extra factor of x is needed.

Question 6Mathematical Physics

Using the definition L{f(t)} = ∫₀^∞ e^{−st} f(t) dt, the Laplace transform of t sin(at) is

  1. 2a/(s² + a²)²
  2. a s²/(s² + a²)²
  3. 2as/(s² + a²)² Correct answer.
  4. a/(s² + a²)

Explanation

The multiplication rule gives L{t f(t)} = −F′(s), with F(s) = a/(s² + a²) for sin(at). Differentiating F with respect to s produces −2as/(s² + a²)², so the transform of t sin(at) is 2as/(s² + a²)².

Question 7Mathematical Physics

The number of inequivalent irreducible representations of the symmetric group S₃ is

  1. 3 Correct answer.
  2. 4
  3. 2
  4. 6

Explanation

For a finite group the number of inequivalent irreducible representations equals the number of conjugacy classes. The classes of S₃ are {e}, the three transpositions, and the two 3-cycles, so there are 3 irreps, of dimensions 1, 1 and 2.

Question 8Mathematical Physics

The extremal of the functional J[y] = ∫₀^{π/2} (y′² − y²) dx satisfying y(0) = 0 and y(π/2) = 1 is

  1. cos x
  2. e^{x} sin x
  3. sinh x/sinh(π/2)
  4. sin x Correct answer.

Explanation

The integrand L = y′² − y² gives the Euler–Lagrange equation y″ + y = 0, whose general solution is A cos x + B sin x. The boundary conditions force A = 0 and B = 1, so the extremal is y = sin x.

Question 9Mathematical Physics

The contour integral ∮ e^{1/z} dz taken counterclockwise around the circle |z| = 2 equals

  1. πi
  2. 2πi e
  3. 0
  4. 2πi Correct answer.

Explanation

Expanding e^{1/z} = 1 + 1/z + 1/(2! z²) + … shows an essential singularity at the origin with residue equal to the coefficient of 1/z, namely 1. The residue theorem then gives 2πi times that residue, so the integral is 2πi.

Question 10Mathematical Physics

The value of the real integral ∫₀^∞ dx/(1 + x⁴) is

  1. π/√2
  2. π/(2√2) Correct answer.
  3. π/4
  4. π/2

Explanation

Closing the contour in the upper half-plane encloses the simple poles at z = e^{iπ/4} and z = e^{3iπ/4}, each contributing a residue of the form e^{−3iπ/4}/(4) up to sign and phase. Summing the residues and applying the residue theorem, then using the evenness of the integrand, yields π/(2√2).

Question 11Mathematical Physics

For a twice continuously differentiable vector field A(x,y,z), the quantity ∇·(∇×A) is equal to

  1. ∇²(∇·A)
  2. |∇×A|²
  3. 0 Correct answer.
  4. ∇·A

Explanation

The curl ∇×A is a vector field, and the divergence of any curl vanishes identically: ∇·(∇×A)=0. This follows from equality of mixed partial derivatives, since the two derivative operators act antisymmetrically.

Question 12Mathematical Physics

The function f(z)=|z|², with z=x+iy, is complex differentiable

  1. at no point of the complex plane
  2. at every point of the complex plane
  3. only at z=0 Correct answer.
  4. only on the unit circle |z|=1

Explanation

Writing f=x²+y² gives u=x²+y² and v=0. The Cauchy–Riemann equations require u_x=v_y and u_y=−v_x, hence 2x=0 and 2y=0, so differentiability can occur only at the origin.

Question 13Mathematical Physics

The Pauli matrix σ_z = diag(1,−1) has eigenvalues

  1. 1 and 2
  2. ±i
  3. ±1 Correct answer.
  4. 0 and 1

Explanation

Since σ_z is diagonal with entries 1 and −1, its eigenvalues are exactly those diagonal entries. Its eigenvectors are the spin-up and spin-down basis states.

Question 14Mathematical Physics

Using the convention f̂(k)=∫_{−∞}^{∞} e^{−ikx} f(x) dx, the Fourier transform of f(x)=e^{−a x²} with a>0 is

  1. √(2πa)e^{−a k²}
  2. (1/a)e^{−k²/a}
  3. πe^{−a k²/2}
  4. √(π/a)e^{−k²/(4a)} Correct answer.

Explanation

Completing the square in the exponent gives ∫e^{−a x²−ikx}dx=e^{−k²/(4a)}∫e^{−a(x+ik/2a)²}dx. The remaining Gaussian integral is √(π/a), so the transform is √(π/a)e^{−k²/(4a)}.

Question 15Mathematical Physics

The Green's function G(x,x′) for G″(x,x′)=δ(x−x′) on 0<x<L with G(0,x′)=G(L,x′)=0 is

  1. x_< (x_> − L)/L Correct answer.
  2. L δ(x−x′)
  3. x_> (x_< − L)/L
  4. x_< x_> / L

Explanation

Continuity at x′ and the unit jump in the derivative determine the two linear pieces. The piecewise form is x_<(x_>−L)/L, which vanishes at both endpoints and has the required jump.

Question 16Mathematical Physics

The eigenvalue of the squared orbital angular momentum operator L² corresponding to the spherical harmonic Y_2^1(θ,φ) is

  1. 2ℏ²
  2. √6 ℏ²
  3. ℏ²
  4. 6ℏ² Correct answer.

Explanation

Spherical harmonics are eigenfunctions of L² with eigenvalue ℏ²l(l+1). For Y_2^1 the orbital quantum number is l=2, so the eigenvalue is 6ℏ².

Question 17Mathematical Physics

In four-dimensional space, the number of independent components of a symmetric rank-2 tensor T_{μν} is

  1. 16
  2. 4
  3. 10 Correct answer.
  4. 6

Explanation

A symmetric rank-2 tensor in n dimensions has n(n+1)/2 independent components because the off-diagonal entries occur in equal pairs. For n=4 this gives 10.

Question 18Mathematical Physics

For a>0 and b>0, the integral ∫_{−∞}^{∞} cos(ax)/(x²+b²) dx equals

  1. (π/b)e^{−ab} Correct answer.
  2. (π/a)e^{−ab}
  3. (π/b)e^{−a/b}
  4. (2π/b)e^{−a/b}

Explanation

Close the contour in the upper half-plane and use e^{iaz}/(z²+b²), whose only enclosed pole is z=ib. The residue there is e^{−ab}/(2ib), so 2πi times the residue gives πe^{−ab}/b.

Question 19Mathematical Physics

For 0<a<1, the integral ∫_0^∞ x^{a−1}/(1+x) dx equals

  1. π/sin(πa) Correct answer.
  2. 2π cos(πa)
  3. π tan(πa)
  4. 1/Γ(a)

Explanation

This is the standard beta integral ∫_0^∞ x^{a−1}/(1+x)dx=B(a,1−a)=Γ(a)Γ(1−a). Euler's reflection formula converts it to π/sin(πa) for 0<a<1.

Question 20Mathematical Physics

For an analytic function f(z)=u(r,θ)+iv(r,θ) expressed in polar coordinates, the Cauchy–Riemann conditions are

  1. u_r = (1/r) v_θ and v_r = −(1/r) u_θ Correct answer.
  2. u_r = r v_θ and v_r = −r u_θ
  3. u_θ = (1/r) v_r and v_θ = −(1/r) u_r
  4. u_r = (1/r) u_θ and v_r = −(1/r) v_θ

Explanation

Writing x=r cosθ and y=r sinθ and applying the chain rule to u_x=v_y and u_y=−v_x gives u_r=(1/r)v_θ and v_r=−(1/r)u_θ. Equivalently, r u_r=v_θ and u_θ=−r v_r. These polar Cauchy–Riemann equations hold wherever f is analytic and r>0.

Question 21Classical Mechanics

A particle moves in a static central potential V(r). Which quantity is conserved specifically because the potential is invariant under rotations about the force centre?

  1. Linear momentum of the particle
  2. Total mechanical energy only when the orbit is circular
  3. Angular momentum about the force centre Correct answer.
  4. Radial component of the particle's velocity

Explanation

Rotational symmetry makes the torque about the force centre zero, so angular momentum about that centre is constant. Energy is also conserved for a static potential, but that follows from time-translation symmetry rather than rotational symmetry. The radial velocity generally changes along a non-circular orbit.

Question 22Classical Mechanics

Which expression gives the moment of inertia of a uniform solid cylinder of mass M and radius R about its symmetry axis?

  1. 2 M R²/3
  2. M R²
  3. 2 M R²/5
  4. M R²/2 Correct answer.

Explanation

For a uniform solid cylinder or disc, the mass is distributed with I = (1/2)MR² about the symmetry axis. The value MR² belongs to a thin cylindrical shell of the same mass and radius, where all the mass lies at radius R.

Question 23Classical Mechanics

A muon has proper lifetime τ0. In a laboratory frame it moves with speed 0.6c. What is its mean lifetime in the laboratory?

  1. 0.8 τ0
  2. τ0
  3. 1.67 τ0
  4. 1.25 τ0 Correct answer.

Explanation

The laboratory lifetime is γτ0 with γ = 1/√(1 − v²/c²). For v = 0.6c, γ = 1/√(1 − 0.36) = 1/0.8 = 1.25, so the laboratory lifetime is 1.25τ0.

Question 24Classical Mechanics

A uniform thin rod of length L is pivoted at one end and performs small oscillations in a vertical plane. What is the angular frequency?

  1. √(3g/(2L)) Correct answer.
  2. √(g/L)
  3. √(g/(2L))
  4. √(2g/(3L))

Explanation

For a physical pendulum, ω = √(mgd/I). For a uniform rod about one end, I = mL²/3 and the centre-of-mass distance is d = L/2, giving ω = √((mgL/2)/(mL²/3)) = √(3g/(2L)).

Question 25Classical Mechanics

A bead of mass m slides on a smooth horizontal circular wire of radius R. Using the angular coordinate θ as the generalized coordinate, which quantity is conserved because θ is cyclic in the Lagrangian?

  1. mR θdot
  2. mR² θdot + mR² θ
  3. mR² θdot²
  4. mR² θdot Correct answer.

Explanation

The Lagrangian is L = (1/2)mR² θdot², which contains no θ. The conjugate momentum pθ = ∂L/∂θdot = mR² θdot is therefore conserved. This is the bead's angular momentum about the vertical axis through the circle centre.

Question 26Classical Mechanics

In Hamiltonian mechanics, if {x,p} = 1, what is the Poisson bracket {x²,p}?

  1. x
  2. 2x Correct answer.
  3. 2xp
  4. 0

Explanation

Use the derivation identity {f,g} = (∂f/∂x)(∂g/∂p) − (∂f/∂p)(∂g/∂x). With f = x² and g = p, the bracket is 2x·1 − 0 = 2x. Equivalently, {x²,p} = 2x{x,p} = 2x.

Question 27Classical Mechanics

Two satellites move in circular orbits around Earth with orbital radii r and 4r. What is the ratio of their orbital periods T(4r)/T(r)?

  1. 4
  2. 8 Correct answer.
  3. 16
  4. 2

Explanation

Kepler's third law for circular orbits gives T² ∝ r³, so T ∝ r^(3/2). Thus T(4r)/T(r) = 4^(3/2) = 8. The mass of the satellite cancels in the circular-orbit relation.

Question 28Classical Mechanics

A uniform solid sphere rolls without slipping down a plane inclined at angle θ to the horizontal. What is its centre-of-mass acceleration along the plane?

  1. g sinθ
  2. 2g sinθ/3
  3. 5g sinθ/7 Correct answer.
  4. 3g sinθ/5

Explanation

Rolling without slipping gives a = αR, and the equation of motion is mg sinθ − f = ma with fR = Iα. For a solid sphere I = (2/5)mR², so a = g sinθ/(1 + 2/5) = 5g sinθ/7.

Question 29Classical Mechanics

A canonical transformation is defined by Q = ln q and P = q p. Which generating function F2(q,P) produces this transformation?

  1. q ln P
  2. P ln q Correct answer.
  3. qP
  4. P/q

Explanation

For a type-2 generating function, p = ∂F2/∂q and Q = ∂F2/∂P. Since p = P/q, integrating p dq at fixed P gives F2 = P ln q plus an arbitrary function of P; the condition Q = ln q fixes that function to be constant. Thus F2(q,P) = P ln q.

Question 30Classical Mechanics

Two equal masses are connected to fixed walls by identical springs and to each other by an identical spring, and they oscillate along the line of the springs. What is the ratio of the lower normal-mode frequency to the higher normal-mode frequency?

  1. 1 : 2
  2. 1 : √3 Correct answer.
  3. 2 : 1
  4. √3 : 1

Explanation

With displacements x1 and x2, the equations are m x1'' = −2kx1 + kx2 and m x2'' = kx1 − 2kx2. The normal-mode eigenvalues are k/m and 3k/m, so the frequencies are √(k/m) and √(3k/m) and their ratio is 1:√3.

Question 31Classical Mechanics

A rigid body moves freely in three-dimensional space with no external constraints. How many degrees of freedom does it have?

  1. Three
  2. Four
  3. Nine
  4. Six Correct answer.

Explanation

A free rigid body has three independent translations of its centre of mass and three independent rotations about its principal axes, giving six generalised coordinates. No constraints are present, so none of these are removed.

Question 32Classical Mechanics

Two Lagrangians for the same mechanical system differ by a term dF(q,t)/dt, where F is a differentiable function. What happens to the equations of motion?

  1. They change by an additive force term.
  2. They are valid only for conservative systems.
  3. They remain unchanged. Correct answer.
  4. They change only if F is quadratic in q.

Explanation

Adding dF(q,t)/dt to a Lagrangian changes the action by boundary terms only, so the Euler–Lagrange equations are identical. The function F need not be quadratic, and the result also applies to systems with velocity-dependent potentials that can be written in Lagrangian form.

Question 33Classical Mechanics

A particle of mass m moves in one dimension under a constant force F in the +x direction. Which Hamiltonian reproduces the correct equation of motion?

  1. H = p²/(2m) + Fx
  2. H = p²/(2m) + Fp
  3. H = p²/(2m) − Fx Correct answer.
  4. H = p²/(2m) + Fx t

Explanation

With H = p²/(2m) − Fx, Hamilton's equations give ẋ = p/m and ṗ = F, so mẍ = F. This corresponds to the potential V = −Fx, which produces a constant force in the +x direction.

Question 34Classical Mechanics

A block slides on a smooth wedge of mass M, and the wedge itself slides freely on a smooth horizontal floor. Gravity acts vertically. In the Lagrangian written with the wedge's horizontal position X and the block's coordinate s along the incline, which coordinate is cyclic?

  1. neither X nor s
  2. s only
  3. X only Correct answer.
  4. both X and s

Explanation

The Lagrangian contains Ẋ but not X itself, because the gravitational potential depends only on the block's position along the incline. Hence X is cyclic and the total horizontal momentum is conserved. The coordinate s is not cyclic because the gravitational potential depends on it.

Question 35Classical Mechanics

A particle of mass m moves in the one-dimensional potential V(x) = V0(x⁴/a⁴ − 2x²/a²), where V0 > 0 and a > 0. What is the angular frequency of small oscillations about either stable minimum at x = ±a?

  1. sqrt(4V0/(m a²))
  2. sqrt(2V0/(m a²))
  3. sqrt(8V0/(m a²)) Correct answer.
  4. sqrt(V0/(m a²))

Explanation

The minima occur at x = ±a. Expanding the potential about either minimum gives V'' = 8V0/a², so the effective spring constant is 8V0/a² and the angular frequency is sqrt(8V0/(m a²)).

Question 36Classical Mechanics

A particle moves in the central potential V(r) = −k/r with k > 0. For a circular orbit of radius r0, what is the angular frequency of the orbital motion?

  1. sqrt(2k/(m r0³))
  2. sqrt(k/(m r0))
  3. sqrt(k/(2m r0³))
  4. sqrt(k/(m r0³)) Correct answer.

Explanation

For a circular orbit, the centripetal force equals the attractive force: m r0 ω² = k/r0². Solving this relation gives ω = sqrt(k/(m r0³)).

Question 37Classical Mechanics

A projectile is launched with speed u at an angle θ above the horizontal over level ground. Neglecting air resistance, what is the radius of curvature of its trajectory at the highest point?

  1. u² cos²θ / g Correct answer.
  2. u² / g
  3. u² sin(2θ) / g
  4. u² sin²θ / g

Explanation

At the highest point the vertical velocity vanishes, so the speed is u cosθ. The acceleration is g downward and is normal to the velocity there, so the radius of curvature is ρ = v²/g = u² cos²θ/g.

Question 38Classical Mechanics

A uniform disk of mass M and radius R lies on a horizontal surface and rolls without slipping. A horizontal force F is applied at its centre. What is the acceleration of the centre of mass?

  1. 3F/(4M)
  2. 2F/(3M) Correct answer.
  3. F/(2M)
  4. F/M

Explanation

Rolling without slipping gives α = a/R. Taking torques about the centre gives fR = (1/2 MR²)(a/R), so the friction force is f = Ma/2. Substituting into F − f = Ma yields a = 2F/(3M).

Question 39Classical Mechanics

A bead of mass m slides without friction on a circular hoop of radius R. The hoop rotates about its vertical diameter with constant angular velocity Ω. Below what critical Ω is the bead's lowest position the only stable equilibrium?

  1. sqrt(g/R) Correct answer.
  2. sqrt(g/(2R))
  3. sqrt(2g/R)
  4. sqrt(R/g)

Explanation

The effective potential is U_eff(θ) = mgR(1 − cosθ) − ½mΩ²R² sin²θ. The lowest position becomes unstable when Ω² exceeds g/R, so the critical angular velocity is sqrt(g/R).

Question 40Classical Mechanics

For a central potential V(r) = −k/r^n with k > 0, circular orbits are stable only when n is less than a certain value. Which listed value of n gives a stable circular orbit?

  1. n = 2
  2. n = 3
  3. n = 1 Correct answer.
  4. n = 4

Explanation

For V = −k/r^n, the second derivative of the effective potential at a circular orbit is proportional to (2 − n). Stability requires n < 2, so among the listed values only n = 1 gives a stable circular orbit.

Question 41Electromagnetic Theory

At the interface between two perfect dielectrics with no free surface charge or current, which component of the electric field is necessarily continuous across the boundary?

  1. Both the normal and tangential components of E
  2. The normal component of E
  3. The tangential component of E Correct answer.
  4. Neither component; only the normal component of D is continuous

Explanation

The boundary condition n × (E2 − E1) = 0 requires the tangential component of E to be continuous across any interface. The normal component of E can jump by σ/ε, so only the tangential part is necessarily continuous when no special surface charge is specified.

Question 42Electromagnetic Theory

A uniform plane wave propagates in a nonmagnetic dielectric with relative permittivity εr = 4. Its phase velocity in the medium is

  1. 2c
  2. c
  3. c/2 Correct answer.
  4. c/4

Explanation

For a nonmagnetic medium μ = μ0 and ε = εr ε0, so the phase velocity is v = 1/√(με) = c/√εr. With εr = 4, this gives v = c/2.

Question 43Electromagnetic Theory

In a linear dielectric medium containing free volume charge density ρf, the differential form of Gauss's law is

  1. ∇·P = −ρf
  2. ∇×D = ρf
  3. ∇·D = ρf Correct answer.
  4. ∇·E = ρf/ε0

Explanation

Since D = ε0E + P, Gauss's law in a dielectric is ∇·D = ρf, where ρf is the free charge density. The bound charge is absorbed into P, while ∇·E involves the total charge density ρf + ρb.

Question 44Electromagnetic Theory

A point charge +q is placed a distance d from an infinite grounded conducting plane. The magnitude of the electrostatic force on the charge is

  1. q²/(32π ε0 d²)
  2. q²/(4π ε0 d²)
  3. q²/(8π ε0 d²)
  4. q²/(16π ε0 d²) Correct answer.

Explanation

The grounded plane is replaced by an image charge −q at distance d behind the plane, so the separation between the real charge and its image is 2d. Coulomb's law then gives F = q²/[4π ε0 (2d)²] = q²/(16π ε0 d²).

Question 45Electromagnetic Theory

A magnetic vector potential in a region is A = (B0/2)(−y x̂ + x ŷ), with B0 constant. The magnetic field B = ∇×A is

  1. zero
  2. −B0 ẑ
  3. B0 ẑ Correct answer.
  4. (B0/2) ẑ

Explanation

Taking the curl gives Bz = ∂Ay/∂x − ∂Ax/∂y = B0/2 − (−B0/2) = B0, while the other components vanish. Hence B = B0 ẑ, a uniform field along z.

Question 46Electromagnetic Theory

The electric field of a plane wave is E = E0[cos(kz − ωt) x̂ + cos(kz − ωt + π/2) ŷ]. The wave is

  1. linearly polarized at 45°
  2. circularly polarized Correct answer.
  3. elliptically polarized with unequal axes
  4. linearly polarized along x

Explanation

The two components have equal amplitudes and a phase difference of π/2, so the electric-field vector traces a circle. Therefore the wave is circularly polarized rather than linear or elliptical with unequal axes.

Question 47Electromagnetic Theory

A rectangular waveguide has broad-wall dimension a = 2.0 cm and narrow-wall dimension b = 1.0 cm, with air filling. The cutoff frequency of the TE10 mode is closest to

  1. 7.50 GHz Correct answer.
  2. 15.0 GHz
  3. 5.00 GHz
  4. 3.75 GHz

Explanation

For the TE10 mode in an air-filled rectangular guide, fc = c/(2a). With a = 2.0 cm, fc = 3×10⁸/(0.04) Hz = 7.5 GHz.

Question 48Electromagnetic Theory

For a time-harmonic plane wave in a lossless medium, the time-averaged Poynting vector expressed in terms of complex phasors E and H is

  1. (1/2) Re(E* × H*)
  2. (1/2) Re(E × H*) Correct answer.
  3. Re(E × H)
  4. (1/2) Re(E × H)

Explanation

For complex phasors E and H, the time average of E(t) × H(t) is (1/2) Re(E × H*). The other expressions either omit the 1/2, use the wrong conjugation, or are not time averages.

Question 49Electromagnetic Theory

An oscillating electric dipole has p(t) = p0 cos(ωt) ẑ in free space. Its time-averaged total radiated power is

  1. p0² ω⁴/(6π ε0 c⁵)
  2. p0² ω⁴/(12π ε0 c³) Correct answer.
  3. p0² ω²/(12π ε0 c³)
  4. p0² ω⁴/(4π ε0 c³)

Explanation

The radiated power follows from the dipole Larmor formula with p̈ = −ω² p0 cos(ωt). Averaging cos² over a cycle gives a factor 1/2, leading to P_avg = p0² ω⁴/(12π ε0 c³).

Question 50Electromagnetic Theory

A point charge moves with constant velocity v. At an instant when its electric field at a field point is E, the magnetic field at that point is given in SI units by

  1. B = (1/c²) v × E Correct answer.
  2. B = 0
  3. B = −(1/c²) v × E
  4. B = (1/c) v × E

Explanation

For a charge moving uniformly, the magnetic field is related to the electric field by B = (1/c²) v × E in SI units. This follows from the Lorentz transformation of the fields and the fact that the charge has no magnetic field in its rest frame.