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Chapter 22: Problem 51

A charge \(+Q\) lies at the origin and \(-3 Q\) at \(x=a .\) Find two points on the \(x\) -axis where \(V=0\)

Short Answer

Expert verified
The two points on x-axis where V=0 are \(x = a\) and \(x = 3a\).

Step by step solution

01

Formulate the equation for total potential V

The total electrostatic potential V at a point P on x-axis is the sum of potentials due to both the charges, i.e. \(V = V1 + V2\). The potential \(V1\) due to charge Q at origin (0,0) at a point P(x,0) on the x-axis is \(V1 = kQ/x\). The potential \(V2\) due to charge -3Q at a(x,0) at the point P(x,0) on the x-axis is \(V2 = -3kQ/(x-a)\). Therefore, the total potential \(V = kQ/x - 3kQ/(x-a)\).
02

Apply the condition for V=0

For V to be 0, the sum of the potentials from the two charges must equal zero. Therefore, we have the equation \(kQ/x = 3kQ/(x-a)\). Simplifying, we get \(x^2 - 3xa = 0\). From this quadratic equation, possible solutions for x are \(x = 0\) and \(x = 3a\).
03

Validate the solutions

The solution \(x = 0\) can be rejected because it is at the location of charge +Q, not a point where V=0. Therefore, the two points on the x-axis where V=0 are \(x = a\) and \(x = 3a\).

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Most popular questions from this chapter

A disk of radius \(a\) carries nonuniform surface charge density \(\sigma=\sigma_{0}(r / a),\) where \(\sigma_{0}\) is a constant. (a) Find the potential at an arbitrary point \(x\) on the disk axis, where \(x=0\) is the disk center. (b) Use the result of (a) to find the electric field on the disk axis, and (c) show that the field reduces to an expected form for \(x \gg a\).

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 \(V=\sigma d / \varepsilon_{0}\)

In considering the potential of an infinite flat sheet, why isn't it useful to take the zero of potential at infinity?

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.

Measurements of the potential at points on the axis of a charged disk are given in the two tables below, one for measurements made close to the disk and the other for measurements made far away. In both tables \(x\) is the coordinate measured along the disk axis with the origin at the disk center, and the zero of potential is taken at infinity. (a) For each set of data, determine a quantity that, when you plot potential against it, should yield a straight line. Make your plots, establish a best-fit line, and determine its slope. Use your slopes to find (b) the total charge on the disk and (c) the disk radius. (Hint: Consult Example 22.7.) Table 1 $$\begin{array}{|l|l|l|l|l|l|} \hline x(\mathrm{mm}) & 2.0 & 4.0 & 6.0 & 8.0 & 10.0 \\ \hline V(\mathrm{V}) & 900 & 876 & 843 & 820 & 797 \\ \hline \end{array}$$ Table 2 $$\begin{array}{|l|l|l|l|l|l|} \hline x(\mathrm{cm}) & 20 & 30 & 40 & 60 & 100 \\ \hline V(\mathrm{V}) & 165 & 118 & 80 & 58 & 30 \\ \hline \end{array}$$
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