To determine the number of excess electrons required, we follow these steps:

Step 1: Use Gauss’s Law

Gauss’s law states that the electric field just outside a conducting sphere is given by:

$E = \frac{Q}{4\pi\epsilon_0 R^2}$

where:

• $E = 5720$ N/C (given electric field)
• $Q$ is the total charge on the sphere
• $R$ is the radius of the sphere
• $\epsilon_0 = 8.85 \times 10^{-12}$ C²/N·m² (permittivity of free space)

Step 2: Solve for $Q$

The radius of the sphere is:

$R = \frac{\text{diameter}}{2} = \frac{17.0 \text{ cm}}{2} = 8.5 \times 10^{-2} \text{ m}$

Rearrange Gauss's law to solve for $Q$:

$Q = 4\pi\epsilon_0 R^2 E$

Substituting values:

$Q = 4\pi (8.85 \times 10^{-12}) (8.5 \times 10^{-2})^2 (5720)$

Let's calculate $Q$. math

Given values

epsilon_0 = 8.85e-12 # C²/N·m² R = 8.5e-2 # meters E = 5720 # N/C

Compute charge Q

Q = 4 * math.pi * epsilon_0 * R**2 * E QThe total charge required on the sphere is approximately $4.60 \times 10^{-9}$ C.

Step 3: Find the Number of Excess Electrons

Since the charge of a single electron is:

$e = 1.602 \times 10^{-19} \text{ C}$

The number of excess electrons is:

$n = \frac{Q}{e} = \frac{4.60 \times 10^{-9}}{1.602 \times 10^{-19}}$

Let's compute $n$. Charge of a single electron e = 1.602e-19 # C

Number of excess electrons

n = Q / e nThe number of excess electrons required is approximately $2.87 \times 10^{10}$ electrons.

Would you like a breakdown of any step?

Related Questions:

1. How would the number of excess electrons change if the electric field were doubled?
2. What would happen to the electric field if the radius of the sphere were increased while keeping the charge the same?
3. How does Gauss’s law apply to a non-conducting sphere instead of a conductor?
4. What would be the surface charge density on the sphere?
5. How does the presence of a nearby conductor affect the charge distribution?

Tip:

In electrostatics, conductors in equilibrium have their excess charge residing entirely on the surface, making Gauss’s law very useful for spherical charge distributions!