Answer
a. The original length of the wire segment is $L_i$, and the final length is $L_f$. The segment has been shortened to 1/3 of its original length, so $L_f$ = $L_i$ / 3. The total charge is Q = 10 nC. The initial linear density is $\lambda_i = Q / L_i$, and the final linear density is $\lambda_f = Q / L_f$. So the relationship between lambdaf and lambdai is $\lambda_f / \lambda_i = (Q / L_f) / (Q / L_i) = L_i / L_f = 3$.
b. The proton is very far from the wire segment, so the electrical force on the proton is practically zero. Therefore, the $F_f/F_i$ ratio is pratically 1.
c. The original length of the wire segment is L_i, and the final length is $L_f = 10L_i$. The total charge $Q = \lambda_i * L_f = \lambda_i * 10L_i$. So the additional charge that needs to be added to the wire segment is $Q - 10 nC = \lambda_i * 10L_i - 10 nC$.
Work Step by Step
a. The original length of the wire segment is $L_i$, and the final length is $L_f$. The segment has been shortened to 1/3 of its original length, so $L_f$ = $L_i$ / 3. The total charge is Q = 10 nC. The initial linear density is $\lambda_i = Q / L_i$, and the final linear density is $\lambda_f = Q / L_f$. So the relationship between lambdaf and lambdai is $\lambda_f / \lambda_i = (Q / L_f) / (Q / L_i) = L_i / L_f = 3$.
b. The proton is very far from the wire segment, so the electrical force on the proton is practically zero. Therefore, the $F_f/F_i$ ratio is pratically 1.
c. The original length of the wire segment is L_i, and the final length is $L_f = 10L_i$. The total charge $Q = \lambda_i * L_f = \lambda_i * 10L_i$. So the additional charge that needs to be added to the wire segment is $Q - 10 nC = \lambda_i * 10L_i - 10 nC$.
