Question

Unreasonable Results

(a) Find the voltage near a \(10.0\;cm\) diameter metal sphere that has \(8.00{\rm{ }}C\) of excess positive charge on it.

(b) What is unreasonable about this result?

(c) Which assumptions are responsible?

Short answer

(a)The voltage near the metal surface is\(1.44 \times {10^{13}}\;V\).

(b)The unreasonable result is: The voltage near the metal sphere is extraordinarily high.

(c) The charge is too much high and diameter is reasonable.

Step-by-step solution

Principle and Formula

Electric Potential Due to a Single Charge: Place a test charge at the desired location,\(q\), and calculate the electric potential energy\({U_{Qq}}\)of the system containing the test charge and the source charge that generates the field.

There is an electric potential of

\(\begin{aligned}V = \frac{{{U_{Qq}}}}{q}\\ = \frac{{kQ}}{r}{\rm{ }}...(1)\end{aligned}\)

Where\(k = 8.99 \times {10^9}\;N\cdot{m^2}/C\), is the constant of Coulomb.

The given data

The diameter of the sphere is:

\(\begin{aligned}D = (10.0\;cm)\left( {\frac{{1\;m}}{{100\;cm}}} \right)\\ = 0.0100\;m.\end{aligned}\)

The radius of the sphere is:

\(\begin{aligned}r = \frac{D}{2}\\ = \frac{{0.0100\;m}}{2}\\ = 0.00500\;m.\end{aligned}\)

The excess charge on the sphere is:\(q = 8.00 C.\)

Calculation of the electric potential or voltage

(a)

The electric potential or the voltage near the metal sphere:

from eq.\((1)\)

\(V = \frac{{kq}}{r}\)

Substituting the given data:

\(\begin{aligned}V = \frac{{\left( {8.99 \times {{10}^9}\;N\cdot{m^2}/C} \right)(8.00{\rm{ }}C)}}{{0.00500\;m}}\\ = 1.44 \times {10^{13}}\;V\end{aligned}\)

Therefore, the voltage near the metal surface is \(1.44 \times {10^{13}}\;V\).

Explaining part (b)

(b)

The voltage near the metal sphere is extraordinarily high, which is unreasonable.

Because the electric field around the metal sphere is too powerful, electrons formed by random ionizations of air molecules around the sphere speed away from their parent molecules.

These fast-moving electrons can ionize more molecules in the vicinity of the sphere, resulting in more free electrons.

With an observable light, the liberated electrons recombine with the ionized air molecules.

Explaining part (c)

The diameter of the sphere is very reasonable but the charge on the sphere is too much and thus responsible for the unreasonable result of part (a).