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Chapter 11: Problem 1613

Two spherical conductors of radii \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\) are at potentials \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) respectively, then what will be the common potential when the conductors are brought in constant? (A) $\left[\left(\mathrm{r}_{1} \mathrm{v}_{1}+\mathrm{r}_{2} \mathrm{v}_{2}\right) /\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right)\right]$ (B) $\left[\left(\mathrm{r}_{1} \mathrm{v}_{1}+\mathrm{r}_{2} \mathrm{v}_{2}\right) /\left(\mathrm{r}_{1}-\mathrm{r}_{2}\right)\right]$ (C) $\left[\left(\mathrm{r}_{1} \mathrm{v}_{1}-\mathrm{r}_{2} \mathrm{v}_{2}\right) /\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right)\right]$ (D) None of these

Short Answer

Expert verified
The common potential when the conductors are brought in contact is given by (A) \(\left[\left(\mathrm{r}_{1} \mathrm{v}_{1}+\mathrm{r}_{2} \mathrm{v}_{2}\right) /\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right)\right]\).

Step by step solution

01

Write the Conservation of Charge Equation

We will write the conservation of charge equation for the problem using the relationship between initial charges and the final charge on the spheres: \(Q_1 + Q_2 = Q_c\)
02

Replace Charges with Capacitance and Potential Differences

We will now substitute the charges in the above equation with capacitance and potential difference expressions: \(C_1 v_1 + C_2 v_2 = C_c v_c\)
03

Replace Capacitance with the Formula for a Spherical Conductor

Now we will replace the capacitance in the above equation with the formula for the capacitance of a spherical conductor (\(C=4\pi \epsilon_0 r\)): \(4\pi \epsilon_0 r_1 v_1 + 4\pi \epsilon_0 r_2 v_2 = 4\pi \epsilon_0 (r_1 + r_2) v_c\)
04

Simplify the Equation and Solve for Common Potential, \(v_c\)

Now we can simplify the equation and solve for the common potential, \(v_c\): \(\frac{r_1 v_1 + r_2 v_2}{r_1 + r_2}= v_c\) Comparing our result with the given options, we find that the answer is: (A) \(\left[\left(\mathrm{r}_{1} \mathrm{v}_{1}+\mathrm{r}_{2} \mathrm{v}_{2}\right) /\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right)\right]\)

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Most popular questions from this chapter

Two Points \(P\) and \(Q\) are maintained at the Potentials of \(10 \mathrm{v}\) and \(-4 \mathrm{v}\), respectively. The work done in moving 100 electrons from \(\mathrm{P}\) to \(\mathrm{Q}\) is \(\ldots \ldots \ldots\) (A) \(2.24 \times 10^{-16} \mathrm{~J}\) (B) \(-9.60 \times 10^{-17} \mathrm{~J}\) (C) \(-2.24 \times 10^{-16} \mathrm{~J}\) (D) \(9.60 \times 10^{-17} \mathrm{~J}\)

The plates of a parallel capacitor are charged up to \(100 \mathrm{~V}\). If $2 \mathrm{~mm}$ thick plate is inserted between the plates, then to maintain the same potential difference, the distance between the capacitor plates is increased by \(1.6 \mathrm{~mm}\) the dielectric constant of the plate is (A) 5 (B) 4 (C) \(1.25\) (D) \(2.5\)

Equal charges \(q\) are placed at the vertices \(A\) and \(B\) of an equilateral triangle \(\mathrm{ABC}\) of side \(\mathrm{a}\). The magnitude of electric field at the point \(c\) is \(\ldots \ldots \ldots\) (A) \(\left(\mathrm{Kq} / \mathrm{a}^{2}\right)\) (B) \(\left.(\sqrt{3} \mathrm{Kq}) / \mathrm{a}^{2}\right)\) (C) \(\left.(\sqrt{2} \mathrm{Kq}) / \mathrm{a}^{2}\right)\) (D) $\left[\mathrm{q} /\left(2 \pi \mathrm{t} \varepsilon_{0} \mathrm{a}^{2}\right)\right]$

Two identical metal plates are given positive charges \(\mathrm{Q}_{1}\) and \(\mathrm{Q}_{2}\left(<\mathrm{Q}_{1}\right)\) respectively. If they are now brought close to gather to form a parallel plate capacitor with capacitance \(\mathrm{c}\), the potential difference between them is (A) \(\left[\left(Q_{1}+Q_{2}\right) /(2 c)\right]\) (B) \(\left[\left(\mathrm{Q}_{1}+\mathrm{Q}_{2}\right) / \mathrm{c}\right]\) (C) \(\left[\left(\mathrm{Q}_{1}-\mathrm{Q}_{2}\right) /(2 \mathrm{c})\right]\) (D) \(\left[\left(\mathrm{Q}_{1}-\mathrm{Q}_{2}\right) / \mathrm{c}\right]\)

A parallel plate air capacitor has a capacitance \(\mathrm{C}\). When it is half filled with a dielectric of dielectric constant 5, the percentage increase in the capacitance will be (A) \(200 \%\) (B) \(33.3 \%\) (C) \(400 \%\) (D) \(66.6 \%\)
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