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Chapter 22: Problem 66

The potential as a function of position in a region is \(V(x)=\) \(3 x-2 x^{2}-x^{3},\) with \(x\) in meters and \(V\) in volts. Find (a) all points on the \(x\) -axis where \(V=0,\) (b) an expression for the electric field, and (c) all points on the \(x\) -axis where \(E=0\)

Short Answer

Expert verified
The potential, V, is zero at \(x=0\) (only real root). The electric field, E, is zero at \(x=0\) and \(x=-2/3\)

Step by step solution

01

Find the roots of potential function

The equation given is \(V(x)=3x-2x^2-x^3\). We first set this equal to zero to find out when the potential is 0. This gives the equation \(0=3x-2x^2-x^3\), or equivalently \(x^3 - 2x^2 + 3x = 0\). The roots of this cubic equation represent the x-values where the potential is zero.
02

Factor the cubic equation

The equation \(x^3 - 2x^2 + 3x = 0\) can be factored by taking an x out of the expression, resulting in \(x*(x^2 - 2x +3) = 0\). Therefore the potential becomes zero when x=0, or when \(x^2 - 2x + 3 = 0\). However, the quadratic equation \(x^2 - 2x +3 =0\) does not have any real roots, so the only real solution for \(V=0\) is at \(x=0\)
03

Find the electric field

The electric field E is the negative derivative of the potential function. This means we differentiate \(V(x)=3x-2x^2-x^3\), with respect to x and then change the sign. This gives us \(E(x)=-\frac{dV}{dx}=2x+3x^2\)
04

Find when electric field is zero

Setting \(E(x)=0\), we get \(2x+3x^2=0\). Factoring out an \(x\) gives \(x*(2+3x)=0\), which implies that E(x)=0, at x=0 and x=-2/3, these points are the solutions to the questions.

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Most popular questions from this chapter

It takes \(45 \mathrm{J}\) to move a 15 -mC charge from point \(A\) to point \(B\). What's the potential difference \(\Delta V_{A B} ?\)

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

What's the charge on an ion that gains \(1.6 \times 10^{-15} \mathrm{J}\) when it moves through a potential difference of \(2500 \mathrm{V} ?\)

The potential in a region is \(V=a x y,\) where \(a\) is a constant. (a) Determine the electric field in the region. (b) Sketch some equipotentials and ficld lines.

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