Lec 11

Magnetic Forces and Magnetic Fields

The Mass Spectrometer

The mass spectrometer is an instrument used to measure the masses and relative abundances of isotopes. Isotopes are atoms with the same number of protons but different numbers of neutrons. In a mass spectrometer, atoms are ionized (typically by removing one electron, so q = +e), accelerated through a potential difference, and then deflected by a magnetic field. The radius of curvature of their path depends on their mass-to-charge ratio, allowing for separation and identification of isotopes.

• Key Equations:

  • Radius of path: $r = \frac{mv}{qB}$

  • Kinetic energy from acceleration: $\frac{1}{2}mv^2 = qV$

  • Combining: $r = \frac{\sqrt{2mV/q}}{B}$

• Application: Used to determine atomic mass and isotope abundance.

Example: The mass spectrum of neon shows three peaks corresponding to isotopes with mass numbers 20, 21, and 22. The height of each peak indicates the relative abundance of each isotope.

The Force on a Current in a Magnetic Field

A magnetic force is exerted on a charged particle moving in a magnetic field, and also on a current-carrying wire. The direction of the force can be determined using the right-hand rule (RHR). The magnitude of the force on a straight wire of length L carrying current I at an angle θ to the magnetic field B is:

• $F = ILB\sin\theta$

• For a single charge: $F = qvB\sin\theta$

Example: A 2.0 m wire carrying 20 A eastward in Earth's 0.5 Gauss (5.0×10-5 T) field (northward) experiences an upward force of $2.0 \times 10^{-3}$ N, about the weight of a sewing pin.

Applications: Magnetohydrodynamic (MHD) Propulsion

MHD propulsion uses the force on a current in a magnetic field to move conductive fluids (like seawater), propelling ships and submarines. Water is expelled by the magnetic force, and by Newton's third law, the ship moves forward.

• Electric power required: $P = IV$

• Force on water: $F = ILB\sin\theta$

• Acceleration: $a = F/m$

Example: For a 1000 kg boat with a 1.0 T field and 100 A current, the force is 200 N, acceleration is 0.2 m/s2, and it takes 50 s to reach 10 m/s. Power required is 2 kW.

Magnetic Fields Produced by Currents

Electric currents produce magnetic fields. The direction of the field around a straight wire can be found using the right-hand rule: thumb in the direction of current, fingers curl in the direction of the magnetic field.

• Magnitude of B-field from a long straight wire: $B = \frac{\mu_0 I}{2\pi r}$

• $\mu_0 = 4\pi \times 10^{-7}$ T·m/A (permeability of free space)

Why must the wire be long? To avoid fringe field effects and ensure the field is uniform and predictable.

Force Between Parallel Currents

Current-carrying wires exert forces on each other. The force per unit length between two parallel wires separated by distance r is:

• $F_{21} = \frac{\mu_0 I_1 I_2 L}{2\pi r}$

• Currents in the same direction attract; opposite directions repel.

Example: Two 1.0 m rods, 0.5 kg each, 7.0 cm apart, can be levitated with a current of 1300 A if the magnetic force balances gravity.

Magnetic Field of a Loop and a Solenoid

The magnetic field at the center of a circular loop of radius R carrying current I is:

• $B = \frac{\mu_0 I}{2R}$

A solenoid is a coil of wire with many turns. The field inside a long solenoid is:

• $B = \mu_0 n I$

• Where n is the number of turns per unit length.

Example: A 0.25 m solenoid with 5000 turns and 3.5 A current produces $B = 0.088$ T (880 G).

Force and Acceleration in a Loudspeaker

A loudspeaker uses the force on a current-carrying coil in a magnetic field to produce sound. The coil moves in response to the current, causing the attached cone to vibrate and create sound waves.

• Force on coil: $F = ILB\sin\theta$

• Acceleration: $a = F/m$

Example: For a coil with 55 turns, 0.025 m diameter, 2.0 A current, and 0.10 T field, the force is 0.86 N and acceleration is 43 m/s2 for a 0.020 kg coil and cone.

The Torque on a Current-Carrying Coil

A current-carrying loop in a magnetic field experiences a torque that tends to align the loop's normal with the field. The net torque on a coil of N turns, area A, and current I in a field B at angle φ is:

• $\tau = NIAB\sin\phi = MB\sin\phi$

• Where $M = NIA$ is the magnetic moment.

Application: Galvanometer

A galvanometer measures current by the rotation of a coil in a magnetic field. The pointer stops when the magnetic torque is balanced by the spring's restoring torque:

• $\tau_M = NIAB\sin\phi$

• $\tau_S = k\Delta\phi$

• At equilibrium: $NIAB\sin\phi = k\Delta\phi$

Application: DC Motor

A DC motor uses the torque on a current-carrying coil to produce rotation. The coil continues to rotate due to inertia even when the current is momentarily zero, and a split-ring commutator reverses the current to maintain rotation.