Topic 4: D. Fields

A field is a region of space where an object experiences a force without being touched. Three field types share the same mathematical skeleton: a source creates the field, and any test object placed in it feels a force. Mastering one field type makes the others feel familiar, which is the whole point of grouping them in Theme D.

The field concept

A field is a way of describing action at a distance. Instead of asking how one mass or charge reaches across empty space to push another, we say the source modifies the space around it, and a second object responds to the field it sits in. Fields are vector quantities: at every point they have a strength and a direction. We visualise them with field lines, which point in the direction a positive test object would be pushed and crowd together where the field is strong. Field lines never cross, because the force at any point has a single direction. Gravitational, electric and magnetic fields all follow this picture, which is why the same toolkit of strength and potential carries across them.

Gravitational fields and Newton's law of gravitation

Every mass attracts every other mass. Newton's law of gravitation states that the force between two point masses is F = G m1 m2 / r squared, where G = 6.67 x 10 to the power minus 11 in SI units and r is the centre-to-centre separation. The force is always attractive and acts along the line joining the masses. Because of the inverse square relationship, doubling the separation cuts the force to one quarter. Worked example: two 1000 kg masses 2.0 m apart attract with F = (6.67e-11)(1000)(1000) / (2.0 squared) = 6.67e-11 / 4 x 1e6 = 1.7 x 10 to the power minus 5 N, a tiny force that shows why gravity only dominates for astronomical masses.

Gravitational field strength, potential and orbits (HL)

Gravitational field strength g is the force per unit mass, g = F / m, measured in N/kg. Around a point mass M, g = G M / r squared. At Earth's surface this gives about 9.8 N/kg, the same number as free-fall acceleration. Gravitational potential V is the work done per unit mass to bring a small mass from infinity to a point, V = minus G M / r in J/kg; it is always negative because gravity is attractive. Potential energy of a mass m is E = m V = minus G M m / r. For orbits, gravity supplies the centripetal force, so G M m / r squared = m v squared / r, giving orbital speed v = square root of (G M / r). Worked example: for a low orbit at r = 6.6e6 m around Earth (M = 6.0e24 kg), v = sqrt(6.67e-11 x 6.0e24 / 6.6e6) = sqrt(6.06e7) = 7.8 x 10 cubed m/s, roughly 7.8 km/s.

Electric fields and Coulomb's law

Charges also act across space. Coulomb's law gives the force between two point charges as F = k q1 q2 / r squared, where k = 8.99 x 10 to the power 9 in SI units. Like charges repel and unlike charges attract, so the sign of the product q1 q2 tells you the direction. Electric field strength E is the force per unit positive charge, E = F / q, measured in N/C (equivalently V/m). Around a point charge, E = k q / r squared, pointing away from a positive charge and toward a negative one. Worked example: a proton (q = 1.6e-19 C) sits 1.0e-10 m from another proton. The force is F = (8.99e9)(1.6e-19 squared) / (1.0e-10 squared) = 8.99e9 x 2.56e-38 / 1.0e-20 = 2.3 x 10 to the power minus 8 N, repulsive.

Electric potential

Electric potential V at a point is the work done per unit positive charge to bring a charge from infinity to that point, V = k q / r, measured in volts (V), where 1 V = 1 J/C. Unlike gravitational potential, electric potential can be positive (near a positive charge) or negative (near a negative charge). The potential energy of a charge q in a potential V is E = q V. In a uniform field, such as between two parallel plates separated by distance d with voltage U across them, the field strength is E = U / d. Worked example: parallel plates 0.020 m apart with 120 V between them produce a uniform field of E = 120 / 0.020 = 6.0 x 10 cubed V/m, or 6.0 x 10 cubed N/C.

Magnetic fields and the force on a current

Magnetic fields are produced by moving charges and by permanent magnets. Field lines run from north to south outside a magnet and form closed loops. A current-carrying wire in a magnetic field experiences a force given by F = B I L sin theta, where B is the magnetic flux density in tesla (T), I is the current, L is the length of wire in the field, and theta is the angle between the wire and the field. The force is greatest when the wire is perpendicular to the field (theta = 90 degrees) and zero when parallel. Fleming's left-hand rule gives the direction: thumb for force, first finger for field, second finger for current. Worked example: a 0.50 m wire carrying 3.0 A perpendicular to a 0.40 T field feels F = (0.40)(3.0)(0.50) = 0.60 N.

The force on a moving charge

Since current is moving charge, a single charge q moving with speed v through a magnetic field B feels a force F = B q v sin theta, where theta is the angle between the velocity and the field. This force is always perpendicular to the velocity, so it changes the direction of motion but never the speed, and therefore does no work on the charge. For a positive charge use the left-hand rule with the second finger pointing along the velocity; for a negative charge, reverse the direction. Worked example: an electron (q = 1.6e-19 C) moving at 2.0e6 m/s perpendicular to a 0.30 T field experiences F = (0.30)(1.6e-19)(2.0e6) = 9.6 x 10 to the power minus 14 N.

Motion of charged particles in fields

Because the magnetic force on a moving charge is always perpendicular to its velocity, a charge entering a uniform magnetic field at right angles travels in a circle. The magnetic force supplies the centripetal force, so B q v = m v squared / r, which rearranges to the radius r = m v / (B q). Faster or heavier particles curve more gently; stronger fields tighten the curve. In a uniform electric field, by contrast, a charge feels a constant force and follows a parabolic path, just like a projectile in gravity. Worked example: a proton (m = 1.67e-27 kg, q = 1.6e-19 C) at 1.0e6 m/s in a 0.50 T field circles with r = (1.67e-27 x 1.0e6) / (0.50 x 1.6e-19) = 1.67e-21 / 8.0e-20 = 2.1 x 10 to the power minus 2 m, about 2.1 cm.

Electromagnetic induction

A changing magnetic field through a circuit creates a voltage, an effect called electromagnetic induction. The key quantity is magnetic flux, defined as the field passing through an area: flux = B A cos theta, measured in webers (Wb), where theta is the angle between the field and the normal to the area. When a conductor of length L moves at speed v perpendicular to a field B, it cuts field lines and a voltage is induced across it, emf = B L v. This is how generators turn motion into electricity. Worked example: a 0.25 m rod moving at 4.0 m/s perpendicular to a 0.20 T field generates emf = (0.20)(0.25)(4.0) = 0.20 V.

Faraday's and Lenz's laws (HL)

Faraday's law states that the induced emf equals the rate of change of magnetic flux linkage: emf = minus N times (change in flux / change in time), where N is the number of turns in a coil. The larger or faster the change in flux, the larger the induced voltage. The minus sign is Lenz's law: the induced current always flows in the direction that opposes the change producing it. This is energy conservation in disguise, since an induced current that aided the change would create energy from nothing. Worked example: a 200-turn coil sees its flux change from 0.030 Wb to 0.010 Wb in 0.040 s. The emf magnitude is N x (change in flux / change in time) = 200 x (0.020 / 0.040) = 200 x 0.50 = 100 V.