Fields and their consequences (A-level only)

Explanation

• A force field is a region in which a body experiences a non-contact force. Gravitational fields arise from masses, electric fields from static charges, and magnetic fields from interactions involving moving charges.
• Field strength is a vector whose direction is found by inspecting the force on a test mass or positive test charge.
• Gravitational and electrostatic fields both obey inverse-square laws and can be represented using field lines, potential and equipotential surfaces.
• Their key difference is that masses always attract, while charges may attract or repel.
• Examiners expect comparisons to distinguish vector field strength from scalar potential and to determine direction from the interaction, not merely from the sign of a potential value.

Explanation

• Newton’s law of gravitation gives the magnitude of the universal attractive force between point masses: $F=Gm_1m_2/r^2$, where $G$ is the gravitational constant and $r$ is their centre-to-centre separation.
• Each mass experiences an equal force directed towards the other mass.
• Outside a spherically symmetric body, its mass may be treated as concentrated at its centre.
• The inverse-square dependence means multiplying separation by a factor $k$ divides force by $k^2$.
• Examiners expect masses in kilograms, separation from centre to centre, a direction as well as a magnitude, and sensible estimation of the extremely small force between ordinary laboratory objects compared with astronomical bodies.

Explanation

• Gravitational field strength is the gravitational force per unit mass on a small test mass, $g=F/m$, with units $\text{N kg}^{-1}$, equivalent to $\text{m s}^{-2}$.
• Field lines point in the direction of force on the test mass.
• Around a spherical mass they are radial and directed inward; their increasing separation with radius represents decreasing strength.
• The magnitude outside the mass is $g=GM/r^2$, where $r$ is measured from the centre.
• Examiners expect altitude to be converted to radial distance, the inward vector direction to be stated, and field-line density to be interpreted qualitatively rather than treated as a literal count of physical lines.

Explanation

• Gravitational potential is work done per unit mass in bringing a small mass from infinity to a point, with zero potential defined at infinity. For a spherical mass, $V=-GM/r$; the negative sign shows that a mass at finite radius is gravitationally bound.
• Potential difference determines external work in a slow transfer, $\Delta W=m\Delta V$.
• No work is done moving along an equipotential surface, which is perpendicular to field lines.
• The radial graphs satisfy $g=-\Delta V/\Delta r$, so field strength is the magnitude of the potential gradient, while potential difference is obtained from the signed area under a $g$–$r$ graph.
• Examiners expect signs, the zero reference and the distinction between scalar potential and vector field strength.

Explanation

• For a circular orbit, gravity supplies the centripetal force, giving $v=\sqrt{GM/r}$. Combining this with $v=2\pi r/T$ gives $T^2=4\pi^2r^3/(GM)$ and therefore $T^2\propto r^3$ for one central mass.
• A circular satellite’s total energy is $-GMm/(2r)$; escape speed follows by setting total energy at infinity to zero. A synchronous satellite matches the body’s rotation period.
• A geostationary satellite is a special synchronous satellite in a circular equatorial orbit, moving in the rotation direction, with orbital radius measured from the planet’s centre.
• Low orbits give shorter periods and different coverage.
• Examiners expect energy, period, speed, orbit plane and radius to be connected to the stated satellite use.

Explanation

• For point charges in a vacuum, Coulomb’s law gives $F=|Q_1Q_2|/(4\pi\varepsilon_0r^2)$, where $\varepsilon_0$ is the permittivity of free space and $r$ is the separation.
• Air may be treated as a vacuum, and a charged sphere’s external effect may be modelled as charge concentrated at its centre.
• Like charges repel and unlike charges attract, so the equation gives magnitude while signs and geometry determine direction.
• Electrostatic and gravitational forces between subatomic particles both follow inverse-square laws, allowing their magnitude ratio to be compared because $r^2$ cancels.
• Examiners expect charge conversion from nanocoulombs, vector addition for several charges, and a direction justified from charge signs.

Explanation

• Electric field strength is force per unit positive test charge, $E=F/Q$, measured in $\text{N C}^{-1}$ or $\text{V m}^{-1}$. Field lines point away from positive charge and towards negative charge.
• A point charge produces radial magnitude $E=Q/(4\pi\varepsilon_0r^2)$.
• Between parallel plates the field is approximately uniform: draw or describe straight, parallel and evenly spaced field lines, and use $E=V/d$, derived from $Fd=Q\Delta V$.
• A charged particle entering perpendicular to a uniform field keeps constant velocity across the field but accelerates parallel or antiparallel to it, giving a parabolic trajectory.
• Examiners expect the field direction to follow force on positive charge, so an electron accelerates opposite to the field, and may require field patterns investigated using conducting paper or an electrolytic tank.

Explanation

• Absolute electric potential is work done per unit positive charge in bringing a small test charge from infinity, where potential is defined as zero. For a point charge, $V=Q/(4\pi\varepsilon_0r)$; potential is a scalar and carries the source charge’s sign.
• External work in a slow transfer is $\Delta W=q\Delta V$.
• No work is done moving along an equipotential surface, which meets field lines at right angles.
• The field points towards decreasing potential and has magnitude given by the potential gradient, while potential difference is the signed area under an $E$–$r$ graph.
• Examiners expect scalar potentials to be added algebraically, vector fields to be added separately, and recognition that zero potential need not mean zero field.

Explanation

• Capacitance is charge stored per unit potential difference, $C=Q/V$, measured in farads, where $1\,\text{F}=1\,\text{C V}^{-1}$.
• The two plates carry equal and opposite charges, and $Q$ denotes the magnitude on either plate rather than a net charge on the whole capacitor.
• For a fixed capacitor, $Q$ is proportional to $V$, so a graph of charge against potential difference is a straight line through the origin with gradient $C$.
• Examiners expect microfarads and nanofarads to be converted to farads, the correct plate-charge interpretation, and a gradient taken from axes in the stated order.

Explanation

• A parallel-plate capacitor has $C=\varepsilon_0\varepsilon_rA/d$, so capacitance increases with overlap area $A$ and relative permittivity $\varepsilon_r$, but decreases with plate separation $d$. In a dielectric, simple polar molecules rotate in the field and the resulting bound surface charges oppose the original field.
• For fixed free charge this reduces potential difference and therefore increases capacitance.
• If the capacitor remains connected to a fixed-voltage supply, extra charge flows onto the plates.
• Experiments may determine relative permittivity or test the $C$–$A$ and $C$–$d$ relationships.
• Examiners expect the overlap area, SI separation and whether charge or voltage is held constant to be identified.

Explanation

• Energy stored by a capacitor is the work done separating charge and is $E=\tfrac12QV=\tfrac12CV^2=Q^2/(2C)$.
• On a graph of potential difference against charge, energy is the area under the line; for a linear capacitor the triangular area gives the factor $1/2$.
• The formula should be selected to match the quantities held fixed.
• If voltage changes, energy released is the difference between initial and final stored energies, not $Q\Delta V$ with one unchanged charge value.
• Examiners expect the squared quantity to be retained, graph axes to be checked, and energy changes to be calculated from two complete energy values.

Explanation

• The time constant of a resistor–capacitor circuit is $\tau=RC$. During discharge, $Q=Q_0e^{-t/RC}$, with corresponding equations for $V$ and current magnitude; after one time constant each is $0.368$ of its initial value and $T_{1/2}=0.69RC$.
• During charging, $Q=Q_0(1-e^{-t/RC})$: charge and voltage rise to limiting values while current falls.
• Graph gradients give rates and area under an $I$–$t$ graph gives transferred charge.
• Required practical 9 includes charge and discharge graphs and a log-linear plot whose gradient is $-1/RC$.
• Examiners expect a time constant from data, not a claim that discharge completes after one $RC$.

Explanation

• A straight current-carrying wire perpendicular to a magnetic field experiences force $F=BIl$.
• Magnetic flux density $B$ is therefore force per unit current per unit length, and one tesla gives $1\,\text{N}$ on a $1\,\text{m}$ wire carrying $1\,\text{A}$ at right angles.
• Fleming’s left-hand rule links field, conventional current and force directions; electron flow is opposite to conventional current.
• Required practical 10 uses a top-pan balance to investigate how force varies with $B$, $I$ and $l$, converting a mass-reading change using $F=\Delta mg$.
• Examiners expect the perpendicular condition, active wire length and mass conversion to be stated.

Explanation

• A charge moving perpendicular to a magnetic field experiences force $F=BQv$, perpendicular to both velocity and field. Fleming’s left-hand rule gives the force on positive conventional current; the direction reverses for a negative particle.
• Since magnetic force is always perpendicular to motion, it does no work and changes direction without changing speed or kinetic energy.
• Equating magnetic and centripetal forces gives $r=mv/(BQ)$ for a circular path.
• This underlies devices such as the cyclotron, whose non-relativistic frequency is $f=BQ/(2\pi m)$ and is independent of orbit radius and speed.
• Examiners expect correct three-dimensional directions, charge magnitude and radius rather than diameter.

Explanation

• Magnetic flux is $\Phi=BA$ when flux density is normal to area $A$, and is measured in webers. For a coil of $N$ turns, flux linkage counts the flux through every turn.
• If $\theta$ is the angle between the field and the coil’s normal, $N\Phi=BAN\cos\theta$. Flux linkage is maximum when the normal is parallel to the field and zero when the coil plane is parallel to the field.
• A negative sign indicates flux opposite to the chosen positive normal.
• Required practical 11 uses a search coil and oscilloscope while varying angle.
• Examiners expect the angle to the normal, not the plane, and clear distinction between flux and flux linkage.