Answer:
The value of R₁/R₂ is 5/9.
Explanation:
CASE I: Effective resistance between A and B
In this situation, there are two cells of resistance "R" connected in series, and this combination is connected in parallel with the series combination other six cells of resistance "R" and "2R". So,
[tex]R_a=R+R=2R[/tex]
[tex]R_b=2R+2R+2R+2R+R+R=10R[/tex]
Ra and Rb are parallelly connected and their effective resistance is R₁.
[tex]\frac{1}{R_1}=\frac{1}{R_a}+\frac{1}{R_b}[/tex] (1)
By substituting the value of Ra and Rb in equation (1) we get;
[tex]\frac{1}{R_1}=\frac{1}{2R}+\frac{1}{10R}=\frac{5+1}{10R}=\frac{6}{10R}=\frac{3}{5R}[/tex]
[tex]R_1=\frac{5R}{3}[/tex] (2)
CASE II: Effective resistance between A and D
In this situation, there are five cells of resistance "R" and "2R" connected in series, and this combination is connected in parallel with the series combination of the other three cells of resistance "2R". So,
[tex]R_c=R+R+R+R+2R=6R[/tex]
[tex]R_d=2R+2R+2R=6R[/tex]
Rc and Rb are parallelly connected and their effective resistance is R₂.
[tex]\frac{1}{R_2}=\frac{1}{R_c}+\frac{1}{R_d}[/tex] (3)
By substituting the value of Rc and Rd in equation (3) we get;
[tex]\frac{1}{R_2}=\frac{1}{6R}+\frac{1}{6R}=\frac{2}{6R}=\frac{1}{3R}[/tex]
[tex]R_2=3R[/tex] (4)
As per the question,
[tex]\frac{R_1}{R_2}=\frac{\frac{5R}{3}}{3R}=\frac{5}{9}[/tex] (5)
Hence, the value of R₁/R₂ is 5/9.