AQA A-Level Physics: Synoptic & Practical Skills

Two apparently unrelated processes - the discharge of a capacitor through a resistor (specification section 3.7, Capacitance) and the radioactive decay of a sample of nuclei (specification section 3.8, Radioactivity / Nuclear physics) - are both described by an exponential law.

A capacitor of capacitance $C = 470\ \mu\text{F}$C=470 μF is charged and then discharged through a resistor of resistance $R = 220\ \text{k}\Omega$R=220 kΩ. A separate radioactive source has a decay constant $\lambda = 9.67 \times 10^{-3}\ \text{s}^{-1}$λ=9.67×10−3 s−1.

Explain why both the charge on the capacitor and the number of undecayed nuclei fall exponentially with time, and link the time constant of the RC circuit to the half-life of the radioactive source. In your answer:

• state the defining (rate) equation for each process and show that each leads to an exponential solution;
• explain why each process has a constant half-life, deriving the relationship $t_{1/2} = RC\ln 2$t1/2​=RCln2 for the capacitor and $t_{1/2} = \dfrac{\ln 2}{\lambda}$t1/2​=λln2​ for the source;
• calculate the half-life of each system and comment on which decays faster.

(9 marks)

In a mass spectrometer, a singly-charged positive ion (charge $+e$+e, where $e = 1.60 \times 10^{-19}\ \text{C}$e=1.60×10−19 C) is first accelerated from rest through a potential difference $V = 4.00\ \text{kV}$V=4.00 kV (specification section 3.2, Particles and radiation / energy). It then enters a uniform magnetic field of flux density $B = 0.350\ \text{T}$B=0.350 T directed perpendicular to its velocity, where it travels in a circular arc (specification sections 3.6 Circular motion and 3.7 Magnetic fields). The radius of the circular path is measured as $r = 0.255\ \text{m}$r=0.255 m.

Explain how this instrument allows the mass of the ion to be determined, and use the data to calculate that mass (in kg and in unified atomic mass units, $1\ \text{u} = 1.66 \times 10^{-27}\ \text{kg}$1 u=1.66×10−27 kg). In your answer you should:

• use the work-energy principle to relate the accelerating pd to the ion's speed;
• explain why the magnetic force provides the centripetal force, and why the speed (and hence the kinetic energy) is unchanged inside the field;
• combine the two results to obtain an expression for the mass in terms of $B$B, $e$e, $r$r and $V$V.

(9 marks)

A student determines the acceleration due to gravity, $g$g, using a simple pendulum. The period is related to the length by $T = 2\pi\sqrt{\frac{L}{g}}, \qquad \text{so} \qquad g = \frac{4\pi^2 L}{T^2}.$T=2πgL​​,sog=T24π2L​.

The student measures the length of the pendulum with a metre rule and times the swings with a hand-operated stopwatch, recording the time for 20 complete oscillations. The measurements are:

Quantity	Value	Uncertainty
Length, $L$L	0.850 m	$\pm 0.002$±0.002 m
Time for 20 oscillations, $t$t	37.0 s	$\pm 0.4$±0.4 s

Calculate a value for $g$g from these data. Then determine the percentage uncertainty in $L$L and in $T$T, combine them correctly (noting that $T$T is squared in the expression for $g$g), and hence quote $g$g with its absolute uncertainty.

(6 marks)

A student determines the resistivity of the metal of a uniform wire. The resistivity is given by $\rho = \frac{RA}{L}, \qquad \text{where the cross-sectional area} \quad A = \frac{\pi d^2}{4}.$ρ=LRA​,where the cross-sectional areaA=4πd2​.

The student measures the resistance $R$R of a length $L$L of the wire and measures its diameter $d$d at several points with a micrometer, taking a mean. The results are:

Quantity	Value	Uncertainty
Resistance, $R$R	4.20 Ω	$\pm 0.05$±0.05 Ω
Length, $L$L	0.750 m	$\pm 0.002$±0.002 m
Diameter, $d$d	$0.38$0.38 mm	$\pm 0.01$±0.01 mm

Calculate the cross-sectional area $A$A and the resistivity $\rho$ρ. Then determine the percentage uncertainty in $R$R, $L$L and $d$d, combine them correctly (noting that $d$d is squared in the area), and hence quote $\rho$ρ with its absolute uncertainty.

(6 marks)

In an AQA Required Practical to determine the acceleration due to gravity by free-fall, a steel ball is held by an electromagnet above a trapdoor switch. When the current to the electromagnet is switched off, the ball is released and an electronic timer starts; the timer stops when the ball strikes and opens the trapdoor switch at the bottom. The student measures the fall height $s$s with a metre rule and the fall time $t$t with the timer, and uses $s = \tfrac{1}{2}gt^2,$s=21​gt2, so that a graph of $s$s against $t^2$t2 has gradient $\tfrac{1}{2}g$21​g.

The student takes a single timing at each of several heights, switches off the electromagnet by hand, and measures each height from the bottom of the ball to the top of the trapdoor.

Evaluate this experimental method. In your answer identify the independent, dependent and control variables, distinguish random from systematic error, comment on reliability, and suggest improvements that would genuinely reduce the uncertainty (and explain why each works).

(6 marks)

Three different situations met in the A-Level course are all described by the same mathematical form, $x = x_0 e^{-kt}:$x=x0​e−kt:

• the charge on a discharging capacitor (specification section 3.7);
• the number of undecayed nuclei in a radioactive sample (specification section 3.8);
• the amplitude of a lightly-damped oscillation (specification section 3.6, Simple harmonic motion).

Explain why all three follow this same exponential form, and state what the constant $k$k represents in each case.

(5 marks)