Understanding Electric Field and Charged Spheres

The question asks us to find the net charge on a conducting sphere, given the electric field strength and direction at a certain distance from its center. For a uniformly charged sphere or a conducting sphere (where charge resides on the surface), the electric field outside the sphere behaves as if all the charge is concentrated at its center.

Key Concepts for Calculating Sphere Charge

• Electric Field (\(E\)): This is the force per unit charge experienced by a test charge placed in the field. Its unit is Newtons per Coulomb (NC\(^{-1}\)).
• Electric Field of a Point Charge or Sphere: At a distance \(r\) from a point charge \(q\) (or the center of a uniformly charged sphere/conducting sphere outside its surface), the magnitude of the electric field is given by Coulomb's law: \(E = \frac{k|q|}{r^2}\).
• Coulomb's Constant (\(k\)): This is a fundamental constant in electrostatics, approximately equal to \(9 \times 10^9\) Nm\(^2\)/C\(^2\).
• Direction of Electric Field: The electric field lines point radially outwards from a positive charge and radially inwards towards a negative charge.

Step-by-Step Calculation of Net Charge

We are given the following information:

• Electric field magnitude, \(E = 1.2 \times 10^3\) NC\(^{-1}\).
• Distance from the center of the sphere, \(r = 20\) cm. We need to convert this to meters: \(r = 20 \text{ cm} = 0.20\) m.
• The electric field points radially inwards. This tells us the charge on the sphere must be negative.

The formula for the electric field magnitude at a distance \(r\) from the center of a sphere with charge \(q\) is:

\(E = \frac{k|q|}{r^2}\)

We need to find the magnitude of the charge, \(|q|\). We can rearrange the formula to solve for \(|q|\):

\(|q| = \frac{E \cdot r^2}{k}\)

Now, we substitute the given values into the equation:

\(|q| = \frac{(1.2 \times 10^3 \text{ NC}^{-1}) \cdot (0.20 \text{ m})^2}{9 \times 10^9 \text{ Nm}^2/\text{C}^2}\)

Calculate the square of the distance:

\((0.20 \text{ m})^2 = 0.040 \text{ m}^2\)

Substitute this back into the equation:

\(|q| = \frac{(1.2 \times 10^3) \cdot (0.040)}{9 \times 10^9} \text{ C}\)

Calculate the product in the numerator:

\(1.2 \times 10^3 \times 0.040 = 0.048 \times 10^3 = 4.8 \times 10^1\)

So the equation becomes:

\(|q| = \frac{4.8 \times 10^1}{9 \times 10^9} \text{ C}\)

Now, perform the division:

\(|q| = \frac{4.8}{9} \times 10^{1-9} \text{ C}\)

\(|q| \approx 0.5333 \times 10^{-8} \text{ C}\)

To express this in scientific notation with the exponent common in the options, we can write:

\(|q| \approx 5.333 \times 10^{-9} \text{ C}\)

Since the electric field is directed radially inwards, the charge on the sphere must be negative. Therefore, the net charge \(q\) is:

\(q = -5.333 \times 10^{-9} \text{ C}\)

Comparing this value with the given options, the closest value is \(-5.3 \times 10^{-9}\) C.

Resulting Net Charge on the Sphere

Based on the calculation using the given electric field and distance, the net charge on the conducting sphere is approximately \(-5.3 \times 10^{-9}\) C. The inwards direction of the electric field confirms the negative sign of the charge.

Revision Table: Electric Field and Charge

Additional Information on Charged Conductors

For a static charge distribution on a conductor:

• The electric field inside the volume of a conductor is always zero.
• Any net charge on a conductor resides entirely on its surface.
• The electric potential is constant throughout the entire volume of a conductor, including its surface.
• The electric field just outside the surface of a conductor is perpendicular to the surface.

In this problem, we are dealing with a conducting sphere, so the electric field outside behaves like that of a point charge located at the center, with the total charge of the sphere.