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16-42P

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Found in: Page 443

### Physics Principles with Applications

Book edition 7th
Author(s) Douglas C. Giancoli
Pages 978 pages
ISBN 978-0321625922

# (II) The field just outside a 3.50-cm-radius metal ball is $${\bf{E = 3}}{\bf{.75 \times 1}}{{\bf{0}}^{\bf{2}}}\;{\bf{N/C}}$$ and points toward the ball. What charge resides on the ball?

The charge resides on the metal ball is $$- 5.11 \times {10^{ - 11}}\;{\rm{C}}$$.

See the step by step solution

## Understanding the electric field

The value of electric field (E) due to a charge Q at any point is determined by finding the electric force (F) per unit charge acting on a small positive test charge (q) placed at that point.

The expression for electric field is given as:

$$E = \frac{F}{q} = k\frac{Q}{{{r^2}}}$$

Here, k is the electrostatic force constant whose value is $$9.0 \times {10^9}\;{\rm{N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}$$.

## Given information:

The electric field outside the metal ball is, $$E = 3.75 \times {10^2}\;{\rm{N/C}}$$

The radius of the metal ball is, $$r = 3.50\;{\rm{cm}} = 3.50 \times 1{{\rm{0}}^{ - 2}}\;{\rm{m}}$$

## Determination of the charge that resides on the ball

If E is the electric field due to charge placed inside the metal ball at its center, then the magnitude of electric field due to the charge Q placed at a distance r from it is:

$$E = k\frac{Q}{{{r^2}}}$$

So, the charge inside the metal ball is given as:

$$Q = \frac{{E{r^2}}}{k}$$

Substitute the values in the above expression.

\begin{aligned}{c}Q = \frac{{\left( {3.75 \times {{10}^2}\;{\rm{N/C}}} \right) \times {{\left( {3.50 \times 1{{\rm{0}}^{ - 2}}\;{\rm{m}}} \right)}^2}}}{{\left( {9.0 \times {{10}^9}\;{\rm{N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right)}}\\ = 5.11 \times {10^{ - 11}}\;{\rm{C}}\end{aligned}

Since any charge in a metallic conductor resides on the surface of the conductor, the magnitude of charge that resides on the metal ball is $$5.11 \times {10^{ - 11}}\;{\rm{C}}$$. Also, as the electric field points towards the metal ball, the charge on the metal ball must be negative.

Thus, the charge that resides on the surface of the metal ball is $$- 5.11 \times {10^{ - 11}}\;{\rm{C}}$$.