A point charge +Q is held at rest at a point P. Another point charge -q, whose mass is m, moves at a constant velocity v in a circular orbit of radius R1 around P. The work required to increase the radius of revolution of -q from R1 to another orbit R2 is (R2 R1)

Question

A point charge +Q is held at rest at a point P. Another point charge -q, whose mass is m, moves at a constant velocity v in a circular orbit of radius R1 around P. The work required to increase the radius of revolution of -q from R1 to another orbit R2 is (R2 > R1) (A) (Qq/2) ( (1/R2) - (1/R1)) (B)  - (Qq/2) ( (1/R2) - (1/R1)) (C) Qq ( (1/R2) - (1/R1)) (D)  - Qq ( (1/R2) - (1/R1))

Tardigrade Admin · Accepted Answer

The force due to point charge +Q on charge -q is F=(k Q(-q)/R2) Therefore work required to increase the radius of revolution of -q from R1 to R2 is W =-∫ limitsR1R2 F . d R =-∫ limitsR1R2-k (Q q/R2) d R=k Q q[(1/R)]R1R2 =-k Q q[(1/R2)-(1/R1)]

$\frac{Qq}{2} (\frac{1}{R _{2}} - \frac{1}{R _{1}})$

$- \frac{Qq}{2} (\frac{1}{R _{2}} - \frac{1}{R _{1}})$

$Qq (\frac{1}{R _{2}} - \frac{1}{R _{1}})$

$- Qq (\frac{1}{R _{2}} - \frac{1}{R _{1}})$

$2 Qq (\frac{1}{R _{2}} - \frac{1}{R _{1}})$

Q. A point charge +Q is held at rest at a point P. Another point charge −q, whose mass is m, moves at a constant velocity v in a circular orbit of radius R1​ around P. The work required to increase the radius of revolution of −q from R1​ to another orbit R2​ is (R2​>R1​)

2Qq​(R2​1​−R1​1​)

−2Qq​(R2​1​−R1​1​)

Qq(R2​1​−R1​1​)

−Qq(R2​1​−R1​1​)

2Qq(R2​1​−R1​1​)

The force due to point charge +Q on charge −q is F=R2kQ(−q)​ Therefore work required to increase the radius of revolution of −q from R1​ to R2​ is W=−R1​∫R2​​F.dR =−R1​∫R2​​−kR2Qq​dR=kQq[R1​]R1​R2​​ =−kQq[R2​1​−R1​1​]

$\frac{Qq}{2} (\frac{1}{R _{2}} - \frac{1}{R _{1}})$

$- \frac{Qq}{2} (\frac{1}{R _{2}} - \frac{1}{R _{1}})$

$Qq (\frac{1}{R _{2}} - \frac{1}{R _{1}})$

$- Qq (\frac{1}{R _{2}} - \frac{1}{R _{1}})$

$2 Qq (\frac{1}{R _{2}} - \frac{1}{R _{1}})$