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A flat circular coil P carrying a current produces a magnetic field. When a second coil Q is placed with its centre a distance x from the centre of coil P, as shown in Fig. 1.1, an e.m.f. V may be induced in coil Q.
It is suggested that $V$ is related to $x$ by the relationship
$V = V_0 e^{-kx}$
where $V_0$ and $k$ are constants.
Design a laboratory experiment to test the relationship between $V$ and $x$. Explain how your results could be used to determine a value for $k$. 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.
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A student is investigating stationary waves on a stretched elastic cord. A vibrator attached to the cord is connected to a signal generator.
The apparatus is set up as shown in Fig. 2.1.
The mass $M$ attached to the cord is adjusted until resonance is obtained. The number $n$ of antinodes on the stationary wave is recorded.
The experiment is repeated with different masses to obtain different values of $n$.
It is suggested that $M$ and $n$ are related by the equation $$f = \frac{n}{2L} \sqrt{\frac{Mg}{\mu}}$$ where $f$ is the frequency of the vibrator, $g$ is the
acceleration of free fall, $L$ is the length of the elastic cord and $\mu$ is the mass per unit length of the elastic cord.
(a) A graph is plotted of $M$ on the $y$-axis against $\frac{1}{n^2}$ on the $x$-axis.
Determine an expression for the gradient.
(b) Values of $n$ and $M$ are given in Fig. 2.2.
The percentage uncertainty in each value of $M$ is $\pm 10\%$.
$$\begin{array}{ccc} n & M/g & \frac{1}{n^2} \\ 3 & 850 \pm & \\ 4 & 500 \pm & \\ 5 & 300 \pm & \\ 6 & 200 \pm & \\ 7 & 150 \pm & \\ 8 & 100 \pm & \\ \end{array}$$
Calculate and record values of $\frac{1}{n^2}$ in Fig. 2.2. [2]
(c) (i) Plot a graph of $M/g$ against $\frac{1}{n^2}$.
Include error bars for $M$. [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. [2]
(d) (i) Using your answers to (a) and (c)(iii), determine the value of $\mu$. Include an appropriate unit.
Data: $g = 9.81 \text{ ms}^{-2}, L = 1.54 \pm 0.01 \text{ m and } f = 120 \pm 5\text{ Hz}$. [3]
(ii) Determine the percentage uncertainty in $\mu$. [1]
(e) The experiment is repeated using the same cord. The frequency is changed to $180 \pm 5\text{ Hz}$.
Determine the mass $M$ required to produce a wave with two antinodes. Include the absolute uncertainty in your answer. [2]
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