In the figure shown A & B are two charged particles having charges q and – q respectively are placed on a nonconducting fixed horizontal smooth plane. B is fixed and A is attached to a non conducting massless spring of spring constant k. The other end of the spring is fixed. Mass of A is m, A and B are in equilibrium when the distance between them is r. Choose the correct options

Options

A. time period of small oscillation of block A about is means position `=2pisqrt((m)/(k-(q^(2))/(2piin_(0)r^(3))))`
B. time period of small oscillation of block A about is means position `=2pisqrt((m)/(k+(q^(2))/(2piin_(0)r^(3))))`
C. To perform SHM, K must be greater than `((q^(2))/(2piepsilon_(0)r^(3)))`
D. to perform SHM, K must be greater than `((2q^(2))/(piepsilon_(0)r^(3)))`

Explanation

Let `x_(0)=` extension in the spring when A is in equilibrium then
`kx_(0)=(1)/(4piin_(0))(q^(2))/(r^(2))` .(i)
Now let A be shifted be a small distance x towards B. Then the resultant force towards A is,
`F_(res)=k(x_(0)+x)-(q^(2))/(4piin_(0)(r-x)^(2))=k(x_(0)+x)-(q^(2))/(4piin_(0)r^(2))(1-(x^(2))/(r))^(-2)`
`=k(x_(0)+x)-(q^(2))/(4piin_(0)r^(2))(1+(2x)/(r))`: `xlt lt r`: binomial expansion
`=kx-(q^(2))/(2piin_(0)r^(3))x`, using (i)
`F_(res)=(k-(q^(2))/(2piin_(0)r^(3)))x`
`becausesFalphaxthereforeSHM` with `T=2pisqrt((m)/(k-(q^(2))/(2piin_(0)r^(3))))`
For reat T, `kgt(q^(2))/(2piin_(0)r^(3))thereforek_("min")=(q^(2))/(2piin_(0)r^(3))`