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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

The displacement of a charge \(Q\) in the electric field $E^{-}=e_{1} i \wedge+e_{2} j \wedge+e_{3} k \wedge\( is \)r^{-}=a i \wedge+b j \wedge$ The work done is \(\ldots \ldots\) (A) \(Q\left(e_{1}+e_{2}\right) \sqrt{\left(a^{2}+b^{2}\right)}\) (B) \(Q\left[\sqrt{ \left.\left(e_{1}^{2}+e_{2}^{2}\right)\right](a+b)}\right.\) (C) \(Q\left(a e_{1}+b e_{2}\right)\) (D) \(\left.Q \sqrt{[}\left(a e_{1}\right)^{2}+\left(b e_{2}\right)^{2}\right]\)

In Millikan's oil drop experiment an oil drop carrying a charge Q is held stationary by a p.d. \(2400 \mathrm{v}\) between the plates. To keep a drop of half the radius stationary the potential difference had to be made $600 \mathrm{v}$. What is the charge on the second drop? (A) \([(3 Q) / 2]\) (B) \((\mathrm{Q} / 4)\) (C) \(Q\) (D) \((\mathrm{Q} / 2)\)

Let $\mathrm{P}(\mathrm{r})\left[\mathrm{Q} /\left(\pi \mathrm{R}^{4}\right)\right] \mathrm{r}$ be the charge density distribution for a solid sphere of radius \(\mathrm{R}\) and total charge \(\mathrm{Q}\). For a point ' \(\mathrm{P}\) ' inside the sphere at distance \(\mathrm{r}_{1}\) from the centre of the sphere the magnitude of electric field is (A) $\left[\mathrm{Q} /\left(4 \pi \epsilon_{0} \mathrm{r}_{1}^{2}\right)\right]$ (B) $\left[\left(\mathrm{Qr}_{1}^{2}\right) /\left(4 \pi \in{ }_{0} \mathrm{R}^{4}\right)\right]$ (C) $\left[\left(\mathrm{Qr}_{1}^{2}\right) /\left(3 \pi \epsilon_{0} \mathrm{R}^{4}\right)\right]$

The capacitors of capacitance \(4 \mu \mathrm{F}, 6 \mu \mathrm{F}\) and $12 \mu \mathrm{F}$ are connected first in series and then in parallel. What is the ratio of equivalent capacitance in the two cases? (A) \(2: 3\) (B) \(11: 1\) (C) \(1: 11\) (D) \(1: 3\)

A thin spherical conducting shell of radius \(\mathrm{R}\) has a charge q. Another charge \(Q\) is placed at the centre of the shell. The electrostatic potential at a point p a distance \((\mathrm{R} / 2)\) from the centre of the shell is ..... (A) \(\left[(q+Q) /\left(4 \pi \epsilon_{0}\right)\right](2 / R)\) (B) $\left[\left\\{(2 Q) /\left(4 \pi \epsilon_{0} R\right)\right\\}-\left\\{(2 Q) /\left(4 \pi \epsilon_{0} R\right)\right]\right.$ (C) $\left[\left\\{(2 Q) /\left(4 \pi \in_{0} R\right)\right\\}+\left\\{q /\left(4 \pi \epsilon_{0} R\right)\right]\right.$ (D) \(\left[(2 \mathrm{Q}) /\left(4 \pi \epsilon_{0} \mathrm{R}\right)\right]\)
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