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A current-carrying coil produces a magnetic field.
It is suggested that the strength $B$ of the magnetic field at the centre of a flat circular coil is inversely proportional to the radius $r$ of the coil.
Design a laboratory experiment that uses a Hall probe to test the relationship between $B$ and $r$. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to
(a) the procedure to be followed,
(b) the measurements to be taken,
(c) the control of variables,
(d) the analysis of the data,
(e) the safety precautions to be taken.
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A scientist is observing some of the moons orbiting the planet Jupiter.
For six different moons, the scientist records the distance \( r \) from the centre of Jupiter and the period \( T \) of the orbit.
It is suggested that \( T \) and \( r \) are related by the equation
\[ T^2 = kr^3 \]
where \( k \) is a constant.
(a) A graph is plotted of \( \lg \, T \) on the \( y \)-axis against \( \lg \, r \) on the \( x \)-axis. Determine the value of the gradient and express the \( y \)-intercept in terms of \( k \). [1]
(b) Values of \( r \) and \( T \) are given in Fig. 2.2.
Calculate and record values of \( \lg \, (r/m) \) and \( \lg \, (T/s) \) in Fig. 2.2. Include the absolute uncertainties in \( \lg \, (T/s) \). [3]
(c) (i) Plot a graph of \( \lg \, (T/s) \) against \( \lg \, (r/m) \). Include error bars for \( \lg \, (T/s) \). [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 uncertainty in your answer. [2]
(iv) Determine the \( y \)-intercept of the line of best fit. Include the uncertainty in your answer. [2]
(d) The constant \( k \) is given by
\[ k = \frac{4 \pi^2}{GM} \]
where the universal gravitational constant \( G = 6.67 \times 10^{-11} \text{N m}^2 \text{kg}^{-2} \) and \( M \) is the mass of Jupiter.
(i) Using your answer to (c)(iv), determine the value of \( k \). Include the uncertainty in your answer. [2]
(ii) Determine the value of \( M \). [1]
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