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Chapter 23: Problem 4

In which situation is the electric potential the highest? a) at a point \(1 \mathrm{~m}\) from a point charge of \(1 \mathrm{C}\) b) at a point \(1 \mathrm{~m}\) from the center of a uniformly charged spherical shell of radius \(0.5 \mathrm{~m}\) with a total charge of \(1 \mathrm{C}\) c) at a point \(1 \mathrm{~m}\) from the center of a uniformly charged rod of length \(1 \mathrm{~m}\) and with a total charge of \(1 \mathrm{C}\) d) at a point \(2 \mathrm{~m}\) from a point charge of \(2 \mathrm{C}\) e) at a point \(0.5 \mathrm{~m}\) from a point charge of \(0.5 \mathrm{C}\)

Short Answer

Expert verified
Answer: All situations have the same electric potential.

Step by step solution

01

Situation A: Point Charge of \(1 \mathrm{C}\)

In this case, we have a point charge of \(1\mathrm{C}\), and the distance from the charge is \(1\mathrm{~m}\). Using the formula for electric potential, \(V_A = \frac{k(1\mathrm{C})}{1\mathrm{~m}} = k\).
02

Situation B: Uniformly Charged Spherical Shell

In this situation, we have a uniformly charged spherical shell with a total charge of \(1\mathrm{C}\) and the distance from the center is \(1\mathrm{~m}\). Since the shell is uniformly charged, we can also use the formula for electric potential, considering the total charge \(Q\). Thus, \(V_B = \frac{k(1\mathrm{C})}{1\mathrm{~m}} = k\).
03

Situation C: Uniformly Charged Rod

For the uniformly charged rod, we can approximate the electric potential at the point \(1\mathrm{~m}\) from the center of the rod (as it is a uniformly charged rod with a length of \(1\mathrm{~m}\)) by using the point charge formula with total charge \(Q\). So, \(V_C= \frac{k(1\mathrm{C})}{1\mathrm{~m}} = k\).
04

Situation D: Point Charge of \(2 \mathrm{C}\)

In this situation, we have a point charge of \(2\mathrm{C}\), and the distance from the charge is \(2\mathrm{~m}\). Using the formula for electric potential, \(V_D =\frac{k(2\mathrm{C})}{2\mathrm{~m}} = k\).
05

Situation E: Point Charge of \(0.5 \mathrm{C}\)

In this case, we have a point charge of \(0.5\mathrm{C}\), and the distance from the charge is \(0.5\mathrm{~m}\). Using the formula for electric potential, \(V_E = \frac{k(0.5\mathrm{C})}{0.5\mathrm{~m}} = k\). Now, let's compare the electric potentials. We have: \(V_A = k\) ; \(V_B = k\) ; \(V_C = k\) ; \(V_D = k\) ; \(V_E = k\). All the electric potentials are equal to k. Therefore, there is no situation in which the electric potential is the highest, as they are all the same.

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Most popular questions from this chapter

A solid conducting sphere of radius \(R\) is centered about the origin of an \(x y z\) -coordinate system. A total charge \(Q\) is distributed uniformly on the surface of the sphere. Assuming, as usual, that the electric potential is zero at an infinite distance, what is the electric potential at the center of the conducting sphere? a) zero c) \(Q / 2 \pi \epsilon_{0} R\) b) \(Q / \epsilon_{0} R\) d) \(Q / 4 \pi \epsilon_{0} R\)

A ring with charge \(Q\) and radius \(R\) is in the \(y z\) -plane and centered on the origin. What is the electric potential a distance \(x\) above the center of the ring? Derive the electric field from this relationship.

The electric potential energy of a continuous charge distribution can be found in a way similar to that used for systems of point charges in Section \(23.6,\) by breaking the distribution up into suitable pieces. Find the electric potential energy of an arbitrary spherically symmetrical charge distribution, \(\rho(r) .\) Do not assume that \(\rho(r)\) represents a point charge, that it is constant, that it is piecewise-constant, or that it does or does not end at any finite radius, \(r\). Your expression must cover all possibilities. Your expression may include an integral or integrals that cannot be evaluated without knowing the specific form of \(\rho(r) .\) (Hint: A spherical pearl is built up of thin layers of nacre added one by one.)

Two fixed point charges are on the \(x\) -axis. A charge of \(-3.00 \mathrm{mC}\) is located at \(x=+2.00 \mathrm{~m}\) and a charge of \(+5.00 \mathrm{mC}\) is located at \(x=-4.00 \mathrm{~m}\) a) Find the electric potential, \(V(x),\) for an arbitrary point on the \(x\) -axis. b) At what position(s) on the \(x\) -axis is \(V(x)=0 ?\) c) Find \(E(x)\) for an arbitrary point on the \(x\) -axis.

What potential difference is needed to give an alpha particle (composed of 2 protons and 2 neutrons) \(200 \mathrm{keV}\) of kinetic energy?
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