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A student is investigating the motion of magnets falling through a vertical copper pipe as shown in Fig. 1.1.
The student releases a magnet above the copper pipe. The magnet has speed $v$ as it leaves the pipe.
It is suggested that the relationship between $v$ and $B$ is
$$v = v_0 e^{-\lambda B}$$
where $B$ is the magnetic flux density at the poles of the magnet and $v_0$ and $\lambda$ are constants.
Design a laboratory experiment to test the relationship between $v$ and $B$. Explain how your results could be used to determine values of $v_0$ and $\lambda$. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to
• the procedure to be followed,
• the measurements to be taken,
• the control of variables,
• the analysis of the data,
• any safety precautions to be taken.
577
563
502
A student is investigating a circuit containing capacitors.
The capacitors are initially uncharged. A capacitor of capacitance $Y$ is charged by connecting it to a power supply. The charge is then shared with another capacitor of capacitance $C$ connected between the terminals $P$ and $Q$, as shown in Fig. 2.1.
A voltmeter is used to measure the maximum potential difference $V$ between $P$ and $Q$.
The experiment is repeated by adding additional capacitors, each of capacitance $C$, in series between $P$ and $Q$.
The total capacitance $X$ between $P$ and $Q$ may be determined by the equation $X = \frac{C}{n}$ where $n$ is the number of capacitors in series.
It is suggested that $V$ and $X$ are related by the equation $YE = (X + Y) V$ where $E$ is the e.m.f. of the power supply.
(a) A graph is plotted of $\frac{1}{V}$ on the y-axis against $X$ on the x-axis.
Determine expressions for the gradient and $y$-intercept.
gradient = ..........................................................
$y$-intercept = ..........................................................
[1]
(b) Values of $n$ and $V$ are given in Fig. 2.2.
Data: $C = (2.7 \pm 0.4) \times 10^{-3} \text{F}$
[Table_1]
Calculate and record values of $X/10^{-3} \text{F}$ and $\frac{1}{V} \text{V}^{-1}$ in Fig. 2.2.
Include the absolute uncertainties in $X$.
[3]
(c)(i) Plot a graph of $\frac{1}{V} \text{V}^{-1}$ against $X/10^{-3} \text{F}$.
Include error bars for $X$.
[2]
(ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled.
[2]
(iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer.
gradient = ..........................................................
[2]
(iv) Determine the $y$-intercept of the line of best fit. Include the absolute uncertainty in your answer.
$y$-intercept = ..........................................................
[2]
(d)(i) Using your answers to (a), (c)(iii) and (c)(iv), determine the values of $E$ and $Y$. Include an appropriate unit for $Y$.
$E = .............................................................. \text{V}$
$Y = ..............................................................$
[2]
(ii) Determine the percentage uncertainty in $Y$.
percentage uncertainty in $Y = ........................................................... \%$ [1]
575
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586
