Question
Lightning occurs when charge builds up in the atmosphere, creating a potential difference between the ground and the atmosphere.
During a lightning strike there is an average current of \(3.3\times10^{4}\,\mathrm{A}\) for a time of \(2.6\times10^{-5}\,\mathrm{s}\).
(a) Calculate the charge transferred during the lightning strike. (2 marks)
charge = ________________________________________________ \( \mathrm{C} \)
(b) The potential difference between the ground and the atmosphere is \(3.0\times10^{7}\,\mathrm{V}\).
Calculate the average power, in GW, transferred during the lightning strike. (2 marks)
power = ________________________________________________ \( \mathrm{GW} \)
(c) A lightning rod is attached to a tall building to conduct charge safely to the ground. The lightning rod is modelled as a uniform cylindrical copper cable of total length \(95\,\mathrm{m}\) that runs from the ground to the top of the building, as shown in Fig. 3.1.
(i) The resistance of the lightning rod is \(9.6\,\Omega\). The resistivity of copper is \(1.7\times10^{-8}\,\Omega\,\mathrm{m}\).
Determine the radius of the lightning rod. (3 marks)
radius = ________________________________________________ \( \mathrm{m} \)
(ii) The radius of the copper lightning rod is doubled with no change to its length.
State the effect of this change on the resistance of the lightning rod. (1 mark)
____________________________________________________________
(d) A section of the lightning rod of length \(0.12\,\mathrm{m}\) is removed for testing. A tensile stress of \(1.9\times10^{6}\,\mathrm{Pa}\) is applied, as shown in Fig. 3.2.
The section of the rod obeys Hooke’s law. The Young modulus of copper is \(1.3\times10^{11}\,\mathrm{Pa}\).
Calculate the extension of the section. (3 marks)
extension = ________________________________________________ \( \mathrm{m} \)
Syllabus Topic Codes (Cambridge International AS & A Level Physics 9702):
• 6.1: Resistance and resistivity — parts (c)(i) and (c)(ii)
• 6.1: Deformation of solids — part (d)
▶️ Answer/Explanation
(a) Charge transferred [2 marks]
The charge transferred is given by:
\( Q=It \)
\( Q=(3.3\times10^{4})(2.6\times10^{-5}) \)
\( Q=0.858\,\mathrm{C} \)
Answer: \( \boxed{0.86\,\mathrm{C}} \)
(b) Average power transferred [2 marks]
Electrical power is given by:
\( P=VI \)
\( P=(3.0\times10^{7})(3.3\times10^{4}) \)
\( P=9.9\times10^{11}\,\mathrm{W} \)
Since \(1\,\mathrm{GW}=10^{9}\,\mathrm{W}\):
\( P=\frac{9.9\times10^{11}}{10^{9}} \)
Answer: \( \boxed{990\,\mathrm{GW}} \)
(c)(i) Radius of the lightning rod [3 marks]
For a cylindrical conductor:
\( R=\frac{\rho L}{A} \)
The cross-sectional area is:
\( A=\pi r^2 \)
Therefore:
\( 9.6=\frac{(1.7\times10^{-8})(95)}{\pi r^2} \)
Rearranging:
\( r=\sqrt{\frac{(1.7\times10^{-8})(95)}{9.6\pi}} \)
\( r=2.3\times10^{-4}\,\mathrm{m} \)
Answer: \( \boxed{2.3\times10^{-4}\,\mathrm{m}} \)
(c)(ii) Effect of doubling the radius [1 mark]
Since:
\( R=\frac{\rho L}{\pi r^2} \)
Resistance is inversely proportional to the square of the radius:
\( R\propto\frac{1}{r^2} \)
If the radius is doubled, the cross-sectional area becomes four times larger. Therefore, the resistance decreases by a factor of four.
Answer: \( \boxed{\text{Resistance decreases by a factor of 4.}} \)
(d) Extension of the section [3 marks]
Young modulus is defined by:
\( E=\frac{\sigma}{\varepsilon} \)
Therefore:
\( \varepsilon=\frac{\sigma}{E} \)
Using \( \varepsilon=\frac{x}{L} \):
\( x=\frac{\sigma L}{E} \)
\( x=\frac{(1.9\times10^{6})(0.12)}{1.3\times10^{11}} \)
\( x=1.75\times10^{-6}\,\mathrm{m} \)
Answer: \( \boxed{1.8\times10^{-6}\,\mathrm{m}} \)