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Chapter 20: Problem 33

Find the magnitude of the electric field due to a charged ring of radius \(a\) and total charge \(Q\) on the ring axis at distance \(a\) from the ring's center.

Short Answer

Expert verified
The magnitude of the electric field at a point on the axis of the ring, at a distance \(a\) from the center of the ring, is given by \(E = k\frac{Q}{2\sqrt{2}a^2}\), where \(k\) is approximately \(8.99 \times 10^9 N m^2/C^2\).

Step by step solution

01

Writing out the given values

First, let's write out the given values from the problem: \(Q\) is the total charge on the ring, \(a\) is the radius of the ring, and \(a\) is also the distance from the center of the ring to the point on the axis. Hence, we have \(r = a\) and \(x = a\).
02

Substituting into the formula for the electric field

Substitute the values from Step 1 into the formula for the electric field due to a charged ring: \(E = k\frac{Qx}{(x^2+r^2)^\frac{3}{2}}\), which leads to \(E = k\frac{Qa}{(a^2+a^2)^\frac{3}{2}}\). Simplify this expression further to get \(E = k\frac{Qa}{(2a^2)^\frac{3}{2}}\).
03

Simplifying the expression further

In this step, the expression will be simplified further. You'll find that \(E = k\frac{Q}{2\sqrt{2}a^2}\).
04

Using the value of Coulomb鈥檚 constant

The value of Coulomb鈥檚 constant \(k\) is approximately \(8.99 \times 10^9 N m^2/C^2\). Substitute this value into the expression to calculate the electric field.

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Most popular questions from this chapter

In his famous 1909 experiment that demonstrated quantization of electric charge, R. A. Millikan suspended small oil drops in an electric field. With field strength \(20 \mathrm{MN} / \mathrm{C},\) what mass drop can be suspended when the drop carries 10 elementary charges?

Suppose the electron and proton charges differed by one part in one billion. Estimate the net charge on your body, assuming it contains equal numbers of electrons and protons.

A proton moving to the right at \(3.8 \times 10^{5} \mathrm{m} / \mathrm{s}\) enters a region where a \(56-\mathrm{kN} / \mathrm{C}\) electric field points to the left. (a) How far will the proton get before it momentarily stops? (b) Describe its subsequent motion.

Equation 20.3 gives the electric field of a point charge. Does the direction of (a) \(\hat{r}\) or (b) \(\vec{E}\) depend on whether the charge is positive or negative?

A straight wire 10 m long carries \(25 \mu C\) distributed uniformly over its length. (a) What's the line charge density on the wire? Find the electric field strength (b) \(15 \mathrm{cm}\) from the wire axis, not near either end, and (c) 350 m from the wire. Make suitable approximations in both cases.
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