A researcher investigates how different grounding paths affect the time required to…

PHYSICS · Multi-part · A Level · Physics · Electricity · Electricity

Problem

A researcher investigates how different grounding paths affect the time required to neutralize a charged conductor. A small solid conducting sphere of radius

1.5\,\text{cm}

initially carries a charge of

-5.0\,\mu\text{C}

. To discharge the sphere, the researcher can attach one of two conductors between the sphere and Earth: • Path X: a

0.80\,\text{m}

-long copper wire of circular cross-section with diameter

0.50\,\text{mm}

• Path Y: a

0.80\,\text{m}

-long aluminum ribbon

2.0\,\text{cm}

wide and

0.10\,\text{mm}

thick The resistivities of copper and aluminum are

\rho_{\text{Cu}} = 1.7\times10^{-8}\;\Omega\,\text{m}

and

\rho_{\text{Al}} = 2.8\times10^{-8}\;\Omega\,\text{m}

, respectively. For the initial moments after connection, the potential of the sphere relative to Earth may be approximated by

V \approx kQ/r

, where

k=9.0\times10^{9}\;\text{N}\,\text{m}^2\,\text{C}^{-2}

and

r

is the sphere’s radius. Assume the potential remains nearly constant while the first

10\%

of the charge is removed. (a) Derive an algebraic expression for the initial current

I_0

that flows through a grounding path of resistance

R

when the sphere with charge

Q_0

and radius

r

is first connected to Earth. (b) Calculate the resistance of Path X and Path Y, showing all steps. (c) Using your result from part (a), compute the initial current for each path and estimate the time required for each path to remove the first

10\%

of the sphere’s initial charge, assuming the current remains constant during that interval. (d) Discuss two assumptions made in your calculations that could cause your time estimates to differ from the actual discharge times observed in a laboratory setting.

Related theory

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