b
Length of \(AB =10 m\)

For battery \(E _{1}\), balancing length is \(l_{1}\) \(l_{1}=380 cm [\) from end \(A ]\)

For battery \(E _{2}\), balancing length is \(l_{2}\) \(l_{2}=760 cm [\) [from end \(A ]\)

Now, we know that \(\frac{ E _{1}}{ E _{2}}=\frac{l_{1}}{l_{2}}\)

\(\Rightarrow \frac{E_{1}}{E_{2}}=\frac{380}{760}=\frac{1}{2}=\frac{a}{b}\)

\(\therefore a=1 \& b=2\)

\(a =1\)