A thin spherical shell with radius R1=3.00R_1 = 3.00 cm is concentric with a larger thin spherical shell with radius R2=5.00R_2 = 5.00 cm. Both shells are made of insulating material. The smaller shell has charge q1=+6.00q_1 = +6.00 nC distributed uniformly over its surface, and the larger shell has charge q2=−9.00q_2 = -9.00 nC distributed uniformly over its surface. Take the electric potential to be zero at an infinite distance from both shells. What is the electric potential due to the two shells at the following distance from their common center: (i) r=0 r=0; (ii) r=4.00r=4.00 cm; (iii) r=6.00r=6.00 cm?

Ch 23: Electric Potential

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 23: Electric Potential

Problem 25a

Chapter 23, Problem 25a

A thin spherical shell with radius $R sub 1 equals 3.00$ cm is concentric with a larger thin spherical shell with radius $R sub 2 equals 5.00$ cm. Both shells are made of insulating material. The smaller shell has charge $q sub 1 equals positive 6.00$ nC distributed uniformly over its surface, and the larger shell has charge $q_{2} = - 9.00$ nC distributed uniformly over its surface. Take the electric potential to be zero at an infinite distance from both shells. What is the electric potential due to the two shells at the following distance from their common center: (i) $r equals 0$ ; (ii) $r equals 4.00$ cm; (iii) $r equals 6.00$ cm?

Verified step by step guidance

Understand that the electric potential due to a spherical shell at a point outside the shell is equivalent to the potential due to a point charge located at the center of the shell. The potential inside a shell is constant and equal to the potential at the surface.

For part (i), at r=0, both shells contribute to the potential. Use the formula for electric potential due to a point charge: V = $\frac{1}{4\pi\varepsilon_0}$ $\frac{q}{r}$. Since r=0, consider the potential at the center due to each shell separately.

For part (ii), at r=4.00 cm, this point is outside the smaller shell but inside the larger shell. Calculate the potential due to the smaller shell using V = $\frac{1}{4\pi\varepsilon_0}$ $\frac{q_1}{r}$. The potential due to the larger shell is constant inside it and equal to the potential at its surface.

For part (iii), at r=6.00 cm, this point is outside both shells. Calculate the potential due to each shell using V = $\frac{1}{4\pi\varepsilon_0}$ $\frac{q}{r}$ for both q_1 and q_2, and sum them to find the total potential.

Remember to consider the sign of each charge when calculating the potential, as the potential due to a negative charge will be negative. Sum the potentials from both shells to find the total potential at each specified distance.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Potential

Electric potential is the work done per unit charge in bringing a positive test charge from infinity to a point in space. It is a scalar quantity measured in volts and is related to the electric field. For a point outside a charged spherical shell, the potential is equivalent to that of a point charge located at the center of the shell.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. For spherical symmetry, it simplifies the calculation of electric fields and potentials. Inside a uniformly charged shell, the electric field is zero, affecting the potential calculation at points within the shell.

Superposition Principle

The superposition principle states that the total electric potential at a point is the algebraic sum of the potentials due to individual charges. This principle is crucial for calculating the potential at various distances from the center, as it allows the addition of potentials from both shells, considering their respective charges and distances.

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Textbook Question

A very long insulating cylinder of charge of radius $2.50$ cm carries a uniform linear density of $15.0$ nC/m. If you put one probe of a voltmeter at the surface, how far from the surface must the other probe be placed so that the voltmeter reads $175$ V?

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are $4.98$ V and $16.2$ V/m, respectively. (Take $V = 0$ at infinity.) What is the distance to the point charge?

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Textbook Question

An infinitely long line of charge has linear charge density $5.00 \times 10^{- 12}$ C/m. A proton (mass $1.67 \times 10^{- 27}$ kg, charge $+ 1.60 \times 10^{- 19}$ C) is $18.0$ cm from the line and moving directly toward the line at $3.50 times 10 cubed$ m/s. Calculate the proton's initial kinetic energy.

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Textbook Question

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are $4.98$ V and $16.2$ V/m, respectively. (Take $V equals 0$ at infinity.) What is the magnitude of the charge?

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are $4.98$ V and $16.2$ V/m, respectively. (Take $V = 0$ at infinity.) Is the electric field directed toward or away from the point charge?

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Textbook Question

An infinitely long line of charge has linear charge density $5.00$\times$10^{-12}$ C/m. A proton (mass $1.67$\times$10^{-27}$ kg, charge $+1.60$\times$10^{-19}$ C) is $18.0$ cm from the line and moving directly toward the line at $3.50$\times$10^3$ m/s. How close does the proton get to the line of charge?

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