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Chapter 17: Problem 88

Charges of \(-12.0 \mathrm{nC}\) and \(-22.0 \mathrm{nC}\) are separated by $0.700 \mathrm{m} .$ What is the potential midway between the two charges?

Short Answer

Expert verified
Question: Determine the electric potential at the midpoint between two charges of \(-12.0 \, \mathrm{nC}\) and \(-22.0 \, \mathrm{nC}\) separated by a distance of \(0.700 \, \mathrm{m}\). Answer: The electric potential at the midpoint between the two charges is \(-876 \, \mathrm{V}\).

Step by step solution

01

Convert the charges to coulombs

We need to convert the charges from nanocoulombs to coulombs. Recall that 1 nC = \(10^{-9}\) C. \(-12.0 \, \mathrm{nC} = -12.0 \times 10^{-9} \, \mathrm{C}\) \(-22.0 \, \mathrm{nC} = -22.0 \times 10^{-9} \, \mathrm{C}\)
02

Find the distance to the midpoint

The distance between the charges is \(0.700 \, \mathrm{m}\), so the distance to the midpoint is half of that: \(r_{mid} = \frac{0.700}{2} \, \mathrm{m} = 0.350 \, \mathrm{m}\)
03

Find the electric potential due to each charge at the midpoint

We can use the formula \(V = k\frac{q}{r}\), where \(k\) is the electrostatic constant (\(k = 8.99 \times 10^9 \, \mathrm{Nm^2/C^2}\)). For the \(-12.0 \times 10^{-9} \, \mathrm{C}\) charge: \(V_1 = k\frac{q_1}{r_{mid}} = (8.99 \times 10^9 \, \mathrm{Nm^2/C^2})(-12.0 \times 10^{-9} \, \mathrm{C}) / (0.350 \, \mathrm{m})\) For the \(-22.0 \times 10^{-9} \, \mathrm{C}\) charge: \(V_2 = k\frac{q_2}{r_{mid}} = (8.99 \times 10^9 \, \mathrm{Nm^2/C^2})(-22.0 \times 10^{-9} \, \mathrm{C}) / (0.350 \, \mathrm{m})\)
04

Sum the electric potentials to find the total potential at the midpoint

We can simply add the potential due to each charge to find the total potential at the midpoint. \(V_{total} = V_1 + V_2\) Plug in the values we found in Step 3 and perform the calculation. \(V_{total} = (-3.08 \times 10^2 \, \mathrm{V}) + (-5.68 \times 10^2 \, \mathrm{V}) = -8.76 \times 10^2 \, \mathrm{V}\) So, the electric potential at the midpoint between the two charges is \(-876 \, \mathrm{V}\).

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Most popular questions from this chapter

A parallel plate capacitor is connected to a \(12-\mathrm{V}\) battery. While the battery remains connected, the plates are pushed together so the spacing is decreased. What is the effect on (a) the potential difference between the plates? (b) the electric field between the plates? (c) the magnitude of charge on the plates?
A parallel plate capacitor has a capacitance of \(2.0 \mu \mathrm{F}\) and plate separation of \(1.0 \mathrm{mm} .\) (a) How much potential difference can be placed across the capacitor before dielectric breakdown of air occurs \(\left(E_{\max }=3 \times 10^{6} \mathrm{V} / \mathrm{m}\right) ?\) (b) What is the magnitude of the greatest charge the capacitor can store before breakdown?
A parallel plate capacitor has a charge of \(5.5 \times 10^{-7} \mathrm{C}\) on one plate and \(-5.5 \times 10^{-7} \mathrm{C}\) on the other. The distance between the plates is increased by \(50 \%\) while the charge on each plate stays the same. What happens to the energy stored in the capacitor?
A defibrillator is used to restart a person's heart after it stops beating. Energy is delivered to the heart by discharging a capacitor through the body tissues near the heart. If the capacitance of the defibrillator is $9 \mu \mathrm{F}\( and the energy delivered is to be \)300 \mathrm{J},$ to what potential difference must the capacitor be charged?
A spherical conductor of radius \(R\) carries a total charge Q. (a) Show that the magnitude of the electric field just outside the sphere is \(E=\sigma / \epsilon_{0},\) where \(\sigma\) is the charge per unit area on the conductor's surface. (b) Construct an argument to show why the electric field at a point \(P\) just outside any conductor in electrostatic equilibrium has magnitude \(E=\sigma / \epsilon_{0},\) where \(\sigma\) is the local surface charge density. [Hint: Consider a tiny area of an arbitrary conductor and compare it to an area of the same size on a spherical conductor with the same charge density. Think about the number of field lines starting or ending on the two areas.]
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