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Chapter 23: Problem 64

A classroom Van de Graaff generator accumulates a charge of \(1.00 \cdot 10^{-6} \mathrm{C}\) on its spherical conductor, which has a radius of \(10.0 \mathrm{~cm}\) and stands on an insulating column. Neglecting the effects of the generator base or any other objects or fields, find the potential at the surface of the sphere. Assume that the potential is zero at infinity.

Short Answer

Expert verified
Question: Determine the potential at the surface of a uniformly charged sphere with a charge of \(1.00 \cdot 10^{-6} \mathrm{C}\) and a radius of \(10.0 \mathrm{~cm}\). Answer: The potential at the surface of the sphere is \(8,990 \mathrm{V}\).

Step by step solution

01

Identify the given information

We are given the following information: - Charge of the sphere, \(Q = 1.00 \cdot 10^{-6} \mathrm{C}\) - Radius of the sphere, \(r = 10.0 \mathrm{~cm} = 0.1 \mathrm{~m}\) (converted to meters)
02

Calculate potential using the formula

We can now calculate the potential at the surface of the sphere using the formula: $$V = \dfrac{kQ}{r}$$ To do so, we will use the value for \(k\) (Coulomb's constant): $$k = 8.99 \cdot 10^{9} \mathrm{\dfrac{N \cdot m^2}{C^2}}$$ Now we can plug in our values: $$V = \dfrac{(8.99 \cdot 10^{9} \mathrm{\dfrac{N \cdot m^2}{C^2}})(1.00 \cdot 10^{-6} \mathrm{C})}{0.1 \mathrm{~m}}$$
03

Solve for the potential

Now we can solve the equation to find the potential at the surface of the sphere: $$V = \dfrac{(8.99 \cdot 10^{9})(1.00 \cdot 10^{-6})}{0.1}$$ $$V = (8.99 \cdot 10^{3})$$ $$V = 8,990 \mathrm{\dfrac{N \cdot m^2}{C^2}}$$ The potential at the surface of the sphere is \(\mathbf{8,990 \mathrm{V}}\).

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Most popular questions from this chapter

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.)

A deuterium ion and a tritium ion each have charge \(+e .\) What work is necessary to be done on the deuterium ion in order to bring it within \(10^{-14} \mathrm{~m}\) of the tritium ion? This is the distance within which the two ions can fuse, as a result of strong nuclear interactions that overcome electrostatic repulsion, to produce a helium-5 nucleus. Express the work in electron-volts.

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

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) \(Q / 4 \pi \epsilon_{0} R\)

Can two equipotential lines cross? Why or why not?
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