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LearnAQA7408First teaching 2015Paper 2 · Paper 2Fields and their consequences

Paper 2 · Module

Fields and their consequences

13 subtopics≈ 465 min in total

Start the module: Gravitational force and field strength Open graphing calculator

Gravitational force and field strength
Every pair of point masses attracts with a force $F=\dfrac{GMm}{r^2}$ , where $G=6.67\times10^{-11}$ N m² kg⁻². A uniform sphere acts as if its mass were at its centre. Gravitational field strength is force per unit mass, $g=\dfrac Fm=\dfrac{GM}{r^2}$ , an inverse-square law. Near Earth's surface, $g$ is nearly constant because height changes are tiny compared with Earth's radius.
≈ 35 min
Electric fields and uniform fields
An electric field is a region where a charge experiences a force. Field strength is the force per unit positive charge: $E=\dfrac Fq$ (N C⁻¹ or V m⁻¹). Between parallel plates with p.d. $V$ and separation $d$ , the field is uniform: $E=\dfrac Vd$ . A charged particle in a uniform field feels a constant force $qE$ , so it accelerates uniformly and, if it enters at right angles, follows a parabola like a projectile.
≈ 35 min
Capacitors and capacitance
A capacitor stores separated charge: $+Q$ on one plate, $-Q$ on the other. Its capacitance is the charge stored per unit p.d.: $C=\dfrac QV$ , measured in farads (F). An isolated sphere also has capacitance, $C=4\pi\varepsilon_0r$ . Capacitors in parallel add: $C=C_1+C_2$ . In series, reciprocals add: $\dfrac1C=\dfrac1{C_1}+\dfrac1{C_2}$ , the opposite of resistors.
≈ 35 min
Characteristics of alternating currents
An alternating current reverses direction periodically: $I=I_0\sin\omega t$ , with peak value $I_0$ , period $T$ and frequency $f=\dfrac1T$ . The power in a resistor, $I^2R$ , varies between $0$ and $I_0^2R$ , so the mean power is $\tfrac12I_0^2R$ . The r.m.s. value is the steady d.c. value giving the same mean power: $I_{\text{rms}}=\dfrac{I_0}{\sqrt2}$ and $V_{\text{rms}}=\dfrac{V_0}{\sqrt2}$ . Pakistan's mains is $230$ V r.m.s., peaking at about $325$ V.
≈ 30 min
Force on a current-carrying conductor
A magnetic field is produced by moving charges or permanent magnets. A wire carrying current $I$ in a field of flux density $B$ feels a force $F=BIL\sin\theta$ , where $\theta$ is the angle between the wire and the field. The force is greatest when they're perpendicular and zero when parallel. Its direction is given by Fleming's left-hand rule. Magnetic flux density is defined as the force per unit current per unit length on a wire at right angles to the field; $1$ T $=1$ N A⁻¹ m⁻¹.
≈ 35 min
Gravitational potential and orbits
Gravitational potential is the work done per unit mass bringing a small mass from infinity: $\varphi=-\dfrac{GM}r$ . It is zero at infinity and negative everywhere else, because gravity is attractive. A mass $m$ has potential energy $E_p=-\dfrac{GMm}r$ . For a circular orbit, gravity provides the centripetal force, giving $v^2=\dfrac{GM}r$ and $T^2=\dfrac{4\pi^2}{GM}r^3$ . A geostationary satellite orbits above the equator with a period of $24$ hours.
≈ 40 min
Energy stored in a capacitor
As a capacitor charges, each new bit of charge is pushed onto the plates against a rising p.d., so the p.d.–charge graph is a straight line through the origin. The energy stored is the area under it: $W=\tfrac12QV=\tfrac12CV^2=\dfrac{Q^2}{2C}$ . A battery charging a capacitor supplies $QV$ , but only half is stored; the other half is dissipated in the circuit resistance.
≈ 30 min
Coulomb's law
Two point charges exert forces on each other proportional to the product of the charges and inversely proportional to the square of their separation: $F=\dfrac{Q_1Q_2}{4\pi\varepsilon_0r^2}$ , with $\varepsilon_0=8.85\times10^{-12}$ F m⁻¹. Like charges repel; unlike attract. A point charge (or charged sphere, from outside) creates a radial field $E=\dfrac Q{4\pi\varepsilon_0r^2}$ . Fields from several charges add as vectors.
≈ 35 min
Force on a moving charge
A charge $Q$ moving at speed $v$ through a field $B$ feels $F=BQv\sin\theta$ , always perpendicular to its velocity. So it does no work: the speed stays constant and, in a uniform perpendicular field, the charge moves in a circle of radius $r=\dfrac{mv}{BQ}$ . Crossed electric and magnetic fields let through only charges with $v=\dfrac EB$ (a velocity selector). In a conductor, the magnetic force pushes carriers to one side, creating the Hall voltage $V_H=\dfrac{BI}{ntq}$ .
≈ 40 min
Electric potential
Electric potential at a point is the work done per unit positive charge bringing a small test charge from infinity: $V=\dfrac Q{4\pi\varepsilon_0r}$ . Unlike gravitational potential, it can be positive (near a positive charge) or negative. A charge $q$ at that point has potential energy $E_P=qV=\dfrac{Qq}{4\pi\varepsilon_0r}$ . Field strength is minus the potential gradient, $E=-\dfrac{\Delta V}{\Delta r}$ , so field lines cross equipotentials at right angles.
≈ 40 min
Discharging a capacitor
When a capacitor discharges through a resistor, the current is proportional to the charge remaining, so charge, p.d. and current all decay exponentially: $x=x_0e^{-t/RC}$ . The time constant $\tau=RC$ is the time for any of them to fall to $\dfrac1e$ ( $37\%$ ) of its starting value. A graph of $\ln V$ against $t$ is a straight line with gradient $-\dfrac1{RC}$ .
≈ 40 min
Magnetic fields due to currents
A current produces a magnetic field. Around a straight wire the field lines are concentric circles (right-hand grip rule), weakening with distance. A flat coil's field resembles a short bar magnet; a long solenoid gives a strong, uniform field inside, which a soft iron core greatly increases. Two parallel wires each sit in the other's field, so they exert forces on each other: currents in the same direction attract, opposite directions repel.
≈ 30 min
Electromagnetic induction
Magnetic flux through an area is $\Phi=BA$ (with $B$ perpendicular to $A$ ), measured in webers. For a coil of $N$ turns, flux linkage is $N\Phi$ . Faraday's law: the induced e.m.f. equals the rate of change of flux linkage. Lenz's law: the induced e.m.f. acts to oppose the change that causes it. Together: $E=-\dfrac{\Delta(N\Phi)}{\Delta t}$ . A conductor of length $L$ moving at $v$ through a field $B$ has $E=BLv$ .
≈ 40 min