91Ó°ÊÓ

$$\begin{gathered} \text{Case}\left(1\right): \\ \text{length}{l}_i=l \\ \text{Charge}{Q}_i=10nC \\ \text{Case}\left(2\right): \\ \text{length}{l}_f=\frac{l}{3} \\ \text{Charge}{Q}_f=10nC \\ \text{Ratioofchargedensities}=\frac{{\mathrm{λ}}_f}{{\mathrm{λ}}_i} \end{gathered}$$

$$\begin{gathered} \text{Linear charge density}\mathrm{λ}=\frac{\text{charge}}{\text{length}} \\ \text{RatioofLinearchargedensities}=\frac{{\mathrm{λ}}_f}{{\mathrm{λ}}_i}=\frac{\frac{Q_f}{l_f}}{\frac{Q_i}{l_i}}……\left(1\right) \\ ⇒\frac{{\mathrm{λ}}_f}{{\mathrm{λ}}_i}=\frac{l_i}{l_f}⋯\left(2\right)……\left(2\right) \end{gathered}$$

Substituting the values of given data in $\text{eqn}\left(2\right)i.e.\text{,}\frac{{\mathrm{λ}}_f}{{\mathrm{λ}}_i}=\frac{l_i}{l_f}$

$$\begin{gathered} ⇒\frac{{\mathrm{λ}}_f}{{\mathrm{λ}}_i}=\frac{l}{\frac{1}{3}} \\ ⇒\frac{{\mathrm{λ}}_f}{{\mathrm{λ}}_i}=3 \end{gathered}$$

$$\begin{gathered} \text{Case}\left(1\right): \\ \text{length}{l}_i=l \\ \text{Charge}{Q}_i=10nC \\ Chargeofproton{Q}_p=Q\left(say\right) \\ \text{Case}\left(2\right): \\ \text{length}{l}_f=\frac{l}{3} \\ \text{Charge}{Q}_f=10nC \\ Chargeofproton{Q}_p=Q\left(say\right) \\ Ratioofforces=\frac{F_f}{F_i} \end{gathered}$$

Substituting the values of given data in$$\begin{gathered} eqn\left(2\right)\text{i.e.,}\frac{{\mathrm{λ}}_f}{{\mathrm{λ}}_i}=\frac{l_i}{l_f} \\ Where,Linear\mathrm{charge}\mathrm{density}\mathrm{λ}=\frac{\text{charge}}{\text{length}}……\left(2\right) \\ ⇒\frac{F_f}{F_i}=\frac{l}{\frac{l}{3}} \\ ⇒\frac{F_f}{F_i}=3 \end{gathered}$$

$$\begin{gathered} \text{Case}\left(1\right): \\ \text{length}{l}_i=l \\ \text{Charge}{Q}_i=10nC \\ \text{Case}\left(2\right): \\ \text{length}{l}_f=10l \\ \text{Ratioofchargedensities}=\frac{{\mathrm{λ}}_f}{{\mathrm{λ}}_i}=1 \end{gathered}$$

Substituting the values of given data in $eqn(2$)

$$\begin{gathered} Linearchargedensity\mathrm{λ}=\frac{\text{charge}}{\text{length}} \\ Ratiooflinearchargedensities=\frac{{\mathrm{λ}}_f}{{\mathrm{λ}}_i}=1=\frac{\frac{{\mathrm{ℓ}}_f}{l_f}}{\frac{Q_i}{l_i}}……\left(1\right) \\ ⇒Q_f=\frac{l_2×Q_i}{l_1}……\left(2\right) \\ {Q}_f=\frac{10l×10\mathrm{nC}}{l} \\ ⇒Q_f=100\mathrm{nC} \end{gathered}$$