$(a)$ Charge on sphere $A , q _{ A }=6.5 \times 10^{-7}\, C$

Charge on sphere $B , q _{ B }=6.5 \times 10^{-7} \,C$

Distance between the spheres, $r=50 \,cm =0.5 \,m$ Force of repulsion between the two spheres

$F=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q_{A} q_{B}}{r^{2}}$

Where, $\varepsilon_{0}=$ Permittivity of free space and $\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} \,Nm ^{2} \,C ^{-2}$

Therefore,

$F =\frac{9 \times 10^{9} \times\left(6.5 \times 10^{-7}\right)^{2}}{(0.5)^{2}}$

$=1.52 \times 10^{-2} \,N$

Therefore, the force between the two spheres is $1.52 \times 10^{-2} \,N$

$(b)$ After doubling the charge, Charge on sphere $A , q _{ A }=1.3 \times 10^{-6} \,C$

Charge on sphere $B , q _{ B }=1.3 \times 10^{-6} \,C$

The distance between the spheres is halved.

$\therefore r=\frac{0.5}{2}=0.25\, m$

Force of repulsion between the two spheres,

$F=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q_{A} q_{B}}{r^{2}}$$=\frac{9 \times 10^{9} \times 1.3 \times 10^{-6} \times 1.3 \times 10^{-6}}{(0.25)^{2}}$

$=16 \times 1.52 \times 10^{-2}$

$=0.243 \,N$

Therefore, the force between the two spheres is $0.243 \,N$.