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Chapter 16: Problem 4

A metallic sphere has a charge of \(+4.0 \mathrm{nC} .\) A negatively charged rod has a charge of \(-6.0 \mathrm{nC} .\) When the rod touches the sphere, $8.2 \times 10^{9}$ electrons are transferred. What are the charges of the sphere and the rod now?

Short Answer

Expert verified
The new charge of the sphere is \(2.688 \times 10^{-9} C\), and the new charge of the rod is \(-4.688 \times 10^{-9} C\).

Step by step solution

01

Calculate the charge of one electron

The charge of one electron (e) is approximately -1.6 x 10^{-19} C. **Step 2: Calculate the Charge Transfer**
02

Find the total charge transferred

The number of electrons transferred is \(8.2 \times 10^9\). We can calculate the charge transfer (q) by multiplying the charge of one electron and the number of electrons: q = (8.2 x 10^9) x (-1.6 x 10^{-19} C) = -1.312 x 10^{-9} C **Step 3: Calculate the New Charge of the Sphere**
03

Find the new charge of the sphere

The initial charge of the sphere (q1) is +4.0 nC. The sphere will receive a negative charge of -1.312 x 10^{-9} C. We can calculate the final charge (q1') by adding the charge transfer to the initial charge: q1' = q1 + q = (+4.0 x 10^{-9} C) + (-1.312 x 10^{-9} C) = 2.688 x 10^{-9} C **Step 4: Calculate the New Charge of the Rod**
04

Find the new charge of the rod

The initial charge of the rod (q2) is -6.0 nC. The rod will lose the same charge (-1.312 x 10^{-9} C) transferred to the sphere. We can calculate the new charge (q2') by adding the charge transfer to the initial charge: q2' = q2 - q = (-6.0 x 10^{-9} C) - (-1.312 x 10^{-9} C) = -4.688 x 10^{-9} C **Step 5: Result**
05

State the final charges of the sphere and the rod

The final charge of the sphere is \(q1' = 2.688 \times 10^{-9} C\), and the final charge of the rod is \(q2' = -4.688 \times 10^{-9} C\).

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Most popular questions from this chapter

A parallel-plate capacitor consists of two flat metal plates of area \(A\) separated by a small distance \(d\). The plates are given equal and opposite net charges \(\pm q\) (a) Sketch the field lines and use your sketch to explain why almost all of the charge is on the inner surfaces of the plates. (b) Use Gauss's law to show that the electric field between the plates and away from the edges is $E=q /\left(\epsilon_{0} A\right)=\sigma / \epsilon_{0} \cdot(\mathrm{c})$ Does this agree with or contra- dict the result of Problem \(70 ?\) Explain. (d) Use the principle of superposition and the result of Problem 69 to arrive at this same answer. [Hint: The inner surfaces of the two plates are thin, flat sheets of charge.]
A raindrop inside a thundercloud has charge - \(8 e\). What is the electric force on the raindrop if the electric field at its location (due to other charges in the cloud) has magnitude $2.0 \times 10^{6} \mathrm{N} / \mathrm{C}$ and is directed upward?
A dipole consists of two equal and opposite point charges \((\pm q)\) separated by a distance \(d .\) (a) Write an expression for the electric field at a point \((0, y)\) on the dipole axis. Specify the direction of the field in all four regions: \(y>\frac{1}{2} d, 0<y<\frac{1}{2} d,-\frac{1}{2} d<y<0,\) and \(y<-\frac{1}{2} d .\) (b) At distant points \((|y| \gg d),\) write a simpler, approximate expression for the field. To what power of \(y\) is the field proportional? Does this conflict with Coulomb's law? [Hint: Use the binomial approximation $\left.(1 \pm x)^{n} \approx 1 \pm n x \text { for } x \ll 1 .\right]$
(a) What would the net charges on the Sun and Earth have to be if the electric force instead of the gravitational force were responsible for keeping Earth in its orbit? There are many possible answers, so restrict yourself to the case where the magnitude of the charges is proportional to the masses. (b) If the magnitude of the charges of the proton and electron were not exactly equal, astronomical bodies would have net charges that are approximately proportional to their masses. Could this possibly be an explanation for the Earth's orbit?
An electron traveling horizontally from west to east enters a region where a uniform electric field is directed upward. What is the direction of the electric force exerted on the electron once it has entered the field?
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