Two equal point charges are fixed at x = -a a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the two equal charges be $q$. The potential energy $U$ of charge $Q$ at a position $x$ is given by the sum of potentials due to the two charge

Vedclass · Accepted Answer

$x^2$

Vedclass · Answer

(B) Let the two equal charges be $q$. The potential energy $U$ of charge $Q$ at a position $x$ is given by the sum of potentials due to the two charges: $U(x) = k q Q [\frac{1}{a-x} + \frac{1}{a+x}]$,where $k = \frac{1}{4 \pi \epsilon_0}$.Simplifying the expression: $U(x) = k q Q [\frac{a+x+a-x}{a^2-x^2}] = k q Q [\frac{2a}{a^2-x^2}]$.The potential energy at the origin $(x=0)$ is $U(0) = k q Q [\frac{2a}{a^2}] = \frac{2 k q Q}{a}$.The change in potential energy is $\Delta U = U(x) - U(0) = k q Q [\frac{2a}{a^2-x^2} - \frac{2}{a}] = 2 k q Q a [\frac{1}{a^2-x^2} - \frac{1}{a^2}]$.$\Delta U = 2 k q Q a [\frac{a^2 - (a^2-x^2)}{a^2(a^2-x^2)}] = 2 k q Q a [\frac{x^2}{a^2(a^2-x^2)}]$.For small displacement $x$,$a^2-x^2 \approx a^2$. Thus,$\Delta U \approx 2 k q Q a [\frac{x^2}{a^4}] = \frac{2 k q Q}{a^3} x^2$.Therefore,$\Delta U \propto x^2$.

Two equal point charges are fixed at $x = -a$ and $x = +a$ on the $x$-axis. Another point charge $Q$ is placed at the origin. The change in the electrical potential energy of $Q$ when it is displaced by a small distance $x$ along the $x$-axis is approximately proportional to

Similar Questions

Two simple pendulums whose lengths are $1\,m$ and $121\,cm$ are suspended side by side. Their bobs are pulled together and then released. After how many minimum oscillations of the longer pendulum will the two be in phase again?

Which one of the following ions is the most stable in aqueous solution? (At. No. $Ti = 22, V = 23, Cr = 24, Mn = 25$)

Two particles $P$ and $Q$ start from the origin and execute Simple Harmonic Motion along the $X$-axis with the same amplitude but with periods $3 \ s$ and $6 \ s$ respectively. The ratio of the velocities of $P$ and $Q$ when they meet is

$A$ particle of mass $4 M$ which is initially at rest explodes into three pieces of masses $M$,$M$ and $2 M$. The equal masses move along $X$ and $Y$-axes with velocities $4 ~ms^{-1}$ and $6 ~ms^{-1}$,respectively. The magnitude of the velocity of the heavier mass is

An observer moves towards a stationary source of sound with a speed $1/5^{th}$ of the speed of sound. The wavelength and frequency of the source emitted are $\lambda$ and $f$ respectively. The apparent frequency and wavelength recorded by the observer are respectively:

Vedclass Test Series

Exam Paper Generator

Online Exam Module