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Chapter 22: Problem 45

A thin spherical shell has radius \(R\) and total charge \(Q\) distributed uniformly over its surface. Find the potential at its center.

Short Answer

Expert verified
The potential at the center of the shell is given by \( V = k \cdot \frac{Q}{R}\), where \( k \) is Coulomb's constant, \( Q \) is the total charge and \( R \) is the radius of the shell.

Step by step solution

01

Charge Distribution

It's important to understand that the potential contributed by each tiny charge element located on the spherical shell to the center is exactly the same, since every charge element is situated at the same distance \( R \) from the center of the sphere.
02

Potential Calculation

To find the total potential at the center of the sphere, sum up the potential by all of the little charge elements. Now, for some small charge \( dQ \) located somewhere on the surface of the sphere, the contribution to the potential at the center of the sphere is \( dV = k \cdot \frac{dQ}{R} \).The total potential \( V \) is thus the integral of \( dV \) over the entire sphere, which results into \( V = k \cdot \int \frac{dQ}{R} \), and that leads to \( V = k \cdot \frac{Q}{R} \).

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Most popular questions from this chapter

An electric field is given by \(\vec{E}=E_{0} \hat{\jmath},\) where \(E_{0}\) is a constant. Find the potential as a function of position, taking \(V=0\) at \(y=0\)

An electron passes point \(A\) moving at \(6.5 \mathrm{Mm} / \mathrm{s} .\) At point \(B\) it comes to a stop. Find the potential difference \(\Delta V_{A B}\)

Two flat metal plates are a distance \(d\) apart, where \(d\) is small compared with the plate size. If the plates carry surface charge densities \(\pm \sigma,\) show that the magnitude of the potential difference between them is \(V=\sigma d / \varepsilon_{0}\)

Two 5.0 -cm-diameter conducting spheres are \(8.0 \mathrm{m}\) apart, and each carries \(0.12 \mu \mathrm{C} .\) Determine (a) the potential on each sphere, (b) the field strength at the surface of each sphere, (c) the potential midway between the spheres, and (d) the potential difference between the spheres.

Find the potential as a function of position in the electric field \(\vec{E}=a x \hat{\imath},\) where \(a\) is a constant and where you're taking \(V=0\) at \(x=0\)
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