Moving Charges and Magnetism

A current I flows through a plane circuit made of two concentric semicircular arcs of radii a and b (with b greater than a) joined at their ends by two straight radial segments, forming a closed "annular half-ring" loop, all lying in one plane with a common centre O. What is the magnitude of the magnetic field at O?

1. (μ0I/4)(1/a + 1/b)
2. (μ0I/4)(1/a − 1/b)
3. (μ0I/2)(1/a − 1/b)
4. Zero, since the straight segments and arcs cancel exactly

A proton and an alpha particle are each accelerated in the same cyclotron, which has a fixed magnetic field B and a fixed maximum dee radius R (the particle is ejected once its orbit radius reaches R). Find the ratio of the maximum kinetic energy of the alpha particle to that of the proton. (Take the alpha particle to have charge 2e and mass 4m, where the proton has charge e and mass m.)

1. 4 : 1
2. 2 : 1
3. 1 : 1
4. 1 : 2

A moving coil galvanometer uses a radial magnetic field (produced by curved pole pieces and a soft iron cylindrical core) instead of a simple uniform magnetic field. What is the key advantage of using a radial field, and what would happen to the deflection scale if a uniform field were used instead?

1. A radial field increases sensitivity only; the scale would remain linear either way
2. A radial field ensures the plane of the coil is always parallel to B at every deflection angle, making torque directly proportional to current (linear scale); with a uniform field, torque would depend on sin(theta), giving a non-uniform scale
3. A radial field eliminates the need for a restoring torsion spring
4. A radial field is used only to reduce the coil's resistance, having no effect on the scale

An infinite plane sheet carries a uniform surface current density K (current per unit length, flowing in a fixed direction within the plane). Which statement correctly describes the magnetic field produced by this sheet?

1. The field decreases as 1/r with distance from the sheet, similar to a long straight wire
2. The field is uniform in magnitude (μ0K/2) on each side of the sheet, independent of distance from it, and directed parallel to the sheet but perpendicular to K
3. The field is zero everywhere except exactly on the sheet
4. The field decreases as 1/r² with distance, similar to a magnetic dipole

A charged particle is released from rest in a region where uniform electric field E (along the y-axis) and uniform magnetic field B (along the z-axis, perpendicular to E) both exist simultaneously. The resulting path is a cycloid. What is the average drift speed of the particle along the x-axis, and on what does it depend?

1. qE/(mB), depends on both charge and mass
2. E/B, independent of the charge, mass, and sign of the particle
3. qB/(mE), depends on both charge and mass
4. Zero, because the particle starts from rest and E and B cancel exactly

A moving coil galvanometer has coil resistance G = 100 ohm and gives full-scale deflection for a current Ig = 1 mA. It is to be converted into (i) an ammeter reading up to 1 A, and (ii) separately, a voltmeter reading up to 100 V. Estimate the order of magnitude of the ratio (shunt resistance for the ammeter) / (series resistance for the voltmeter).

1. of order 10⁻³
2. of order 10⁻⁴
3. of order 10⁻⁶
4. of order 10⁻⁸

A planar current loop carrying current I has area A and lies in a uniform magnetic field B, with its plane making an angle theta with B, so the magnetic moment vector makes angle (90 - theta) with B. The loop is now reshaped (still planar, still carrying current I) into a completely different shape but with the same enclosed area A and the same orientation. How does the torque on the loop change?

1. Torque increases because the new shape has a larger perimeter
2. Torque decreases because the current path is no longer symmetric
3. Torque remains exactly the same, since it depends only on m = IA and orientation, not on the loop's shape
4. Torque becomes zero because only circular loops experience torque in a magnetic field

A wire carrying current I is bent into a semicircular arc of radius R lying in the xy-plane, with the two straight ends of the diameter left open (unconnected). A uniform magnetic field B exists everywhere, directed along the z-axis (perpendicular to the plane of the arc). What is the magnitude of the net magnetic force on the semicircular arc?

1. πBIR
2. BIR
3. 2BIR
4. Zero, because the arc is curved and forces cancel by symmetry

A charged particle enters a uniform magnetic field with its velocity making an angle theta with the field direction, resulting in helical motion. Find the ratio of the pitch of the helix to the circumference of the circular cross-section of the helix.

1. tan(theta)
2. cot(theta)
3. sin(theta)
4. cos(theta)

In a simple (fixed-frequency) cyclotron, protons are accelerated to speeds that are a significant fraction of the speed of light. Why does a simple cyclotron fail to further accelerate the protons beyond a certain energy, even though the magnetic field and applied alternating voltage are kept unchanged?

1. The proton's charge decreases at high speed, reducing the force on it
2. The relativistic increase in mass increases the period of revolution, so the proton falls out of phase with the fixed-frequency oscillating voltage
3. The magnetic force becomes zero once the proton reaches relativistic speed
4. The radius of the path becomes independent of speed at high energy, so the proton cannot move outward

A long solenoid (length much greater than radius) has n turns per unit length and carries current I. What is the ratio of the magnetic field measured exactly at one open end of the solenoid (on the axis) to the field at the centre of the solenoid?

1. 1 : 1
2. 1 : 2
3. 1 : 4
4. 2 : 1

An ideal toroid is assumed to have zero magnetic field outside its core because the field lines are entirely confined within the doughnut-shaped winding. In practice, however, the wire must also travel once around the toroidal axis to complete the circuit, so the toroid also behaves like a single circular loop of current I with radius equal to the mean radius R. If the toroid has N turns and carries current I, what is the order of magnitude of the ratio of this "stray" field (at the centre of the equivalent single loop) to the main field inside the toroid?

1. N
2. N/π
3. π/N
4. N²

A long cylindrical wire of radius R carries a current I whose current density varies as J(r) = J0*(r/R), increasing linearly from the axis to the surface. Find the magnetic field at r = R/2.

1. μ0I/(4πR)
2. μ0I/(2πR)
3. μ0I/(8πR)
4. μ0I/(16πR)

Two long straight parallel wires are separated by a distance d. Wire A carries current 2I into the page and wire B carries current I out of the page. At what distance from wire B (measured away from wire A, outside the pair) is the net magnetic field zero?

1. d/3
2. d/2
3. d
4. 2d

A square loop of wire of side "a" carries a current I. A circular loop is made from a wire of exactly the same length (same total perimeter) and carries the same current I. Compare the magnetic field at the centre of the two loops.

1. Field at centre of circular loop is greater, ratio (circle : square) is about 1.15 : 1
2. Field at centre of square loop is greater, ratio (square : circle) is about 1.15 : 1
3. Both fields are exactly equal, since total current and length are the same
4. Field at centre of circular loop is greater, ratio (circle : square) is about 2 : 1