Question:
The work done on a particle of mass $m$ by a force $K \left[\frac{ x }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}} \hat{ i }+\frac{ y }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}} \hat{ j }\right]$ ( $K$ being a constant of appropriate dimensions, when the particle is taken from the point $( a , 0)$ to the point $(0, a )$ along a circular path of radius a about the origin in the $x$-y plane is
The work done on a particle of mass $m$ by a force $K \left[\frac{ x }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}} \hat{ i }+\frac{ y }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}} \hat{ j }\right]$ ( $K$ being a constant of appropriate dimensions, when the particle is taken from the point $( a , 0)$ to the point $(0, a )$ along a circular path of radius a about the origin in the $x$-y plane is
Updated On: Jun 14, 2022
- $\frac{ 2K \pi}{a}$
- $\frac{ K \pi}{a}$
- $\frac{ K \pi}{2a}$
- 0
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The Correct Option is D
Solution and Explanation
$d w =\vec{ F } \cdot d \vec{ r }=\vec{ F } \cdot( d x \hat{ i }+ d \hat{ j }) $
$= K \int \frac{ x dx }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}}+\frac{ ydy }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}}$
$x ^{2}+ y ^{2}= a ^{2}$
$w =\frac{ K }{ a ^{3}} \int\limits_{ a }^{0} x d x +\int\limits_{0}^{ a } ydy $
$=\frac{ K }{ a ^{3}}\left(\frac{- a ^{2}}{2}+\frac{ a ^{2}}{2}\right)=0$
$= K \int \frac{ x dx }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}}+\frac{ ydy }{\left( x ^{2}+ y ^{2}\right)^{3 / 2}}$
$x ^{2}+ y ^{2}= a ^{2}$
$w =\frac{ K }{ a ^{3}} \int\limits_{ a }^{0} x d x +\int\limits_{0}^{ a } ydy $
$=\frac{ K }{ a ^{3}}\left(\frac{- a ^{2}}{2}+\frac{ a ^{2}}{2}\right)=0$
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Concepts Used:
Electrostatic Potential
The electrostatic potential is also known as the electric field potential, electric potential, or potential drop is defined as “The amount of work that is done in order to move a unit charge from a reference point to a specific point inside the field without producing an acceleration.”
SI Unit of Electrostatic Potential:
SI unit of electrostatic potential - volt
Other units - statvolt
Symbol of electrostatic potential - V or φ
Dimensional formula - ML2T3I-1
Electric Potential Formula:
The electric potential energy of the system is given by the following formula:
U = 1/(4πεº) × [q1q2/d]
Where q1 and q2 are the two charges that are separated by the distance d.