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Chapter 22: Problem 25

The potential at the surface of a 10 -cm-radius sphere is \(4.8 \mathrm{kV}\) What's the sphere's total charge, assuming charge is distributed in a spherically symmetric way?

Short Answer

Expert verified
The total charge on the sphere is calculated using Gauss's Law and the given potential and radius. After substituting these given values into the formula and calculating, the sphere's total charge will be found.

Step by step solution

01

Identify all given quantities

From the problem, we know that the radius (r) of the sphere is 10 cm, which is equal to \(0.1 m\) when converted to standard units. And the potential (V) at the surface of the sphere is \(4.8 kV\) or \(4800 V\). The electric constant (蔚), also known as permittivity of free space, is \(8.85 x 10^{-12} C^{2}/N路m^{2}\). Using these values, we aim to find the charge (Q) on the sphere.
02

Use the formula for potential at the surface of a sphere

The formula for potential at the surface of a sphere is \(V = \frac{Q}{4蟺蔚r}\). Here, we are looking to solve for Q, the charge on the sphere. Rearranging the terms, we have \(Q = V \cdot 4蟺蔚r\).
03

Substituting the values into the formula

Substitute the given values into the formula: \(Q = \(4800 V \cdot 4蟺 \cdot 8.85 x 10^{-12} C^{2}/N路m^{2} \cdot 0.1 m\).
04

Calculation

After substituting the values into the formula, calculate the numerical value. Simplify the expression to find the total charge Q.

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Most popular questions from this chapter

A line charge extends along the \(x\) -axis from \(-L / 2\) to \(L / 2 .\) Its line charge density is \(\lambda=\lambda_{0}(x / L)^{2},\) where \(\lambda_{0}\) is a constant. Find an expression for the potential on the \(x\) -axis for \(x > L / 2 .\) Check that your expression reduces to an expected result for \(x \gg L\)

The potential is constant throughout an entire volume. What must be true of the electric field within that volume?

A large metal sphere has three times the diameter of a smaller sphere and carries three times the charge. Both spheres are isolated, so their surface charge densities are uniform. Compare (a) the potentials (relative to infinity) and (b) the electric field strengths at their surfaces.

Show that the result of Example 22.8 approaches the field of a point charge for \(x \gg a .\) (Hint: You'll need to apply the binomial approximation from Appendix A to the expression \(1 / \sqrt{x^{2}+a^{2}}\) )

The electric potential in a region increases linearly with distance. What can you conclude about the electric field in this region?
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