Assuming all bulbs are identical, rank the brightnesses of the bulbs, from brightest to dimmest.

Assuming all bulbs are identical, rank the brightnesses of the bulbs, from brightest to dimmest.

1. A

A > D - H > B - C > E - F = G

2. B

A > B - C > D - H > E - F = G

3. C

A > D - H > E - F - G > B = C

4. D

All have equal brightness

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Solution:

For the branch containing A, B and C.

${\mathrm{R}}_{eq}\text{\hspace{0.17em}}=\frac{3R}{2}$

Current in A : ${\mathrm{I}}_{1}=\frac{\mathrm{E}}{{\mathrm{R}}_{\mathrm{eq}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}\frac{2\mathrm{E}}{3\mathrm{R}}$

Current in each of B and C = ${{I}^{\text{'}}}_{1}=\frac{{I}_{1}}{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}\frac{E}{3R}$

For the branch containing D, E, F, G and H:

${\mathrm{R}}_{eq}\text{\hspace{0.17em}}=\frac{7R}{3}$

Current through D and H: ${I}_{2}=\frac{3E}{7R}$

Current through each of E, F and G:

${{\mathrm{I}}^{\text{'}}}_{1}=\frac{{\mathrm{I}}_{2}}{3}\text{\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}\frac{\mathrm{E}}{7\mathrm{R}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{Since}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{I}}_{1}\text{\hspace{0.17em}\hspace{0.17em}}>\text{\hspace{0.17em}}{\mathrm{I}}_{2}\text{\hspace{0.17em}}>\text{\hspace{0.17em}\hspace{0.17em}}{{\mathrm{I}}^{\text{'}}}_{1}\text{\hspace{0.17em}}>\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{I}}^{\text{'}}$

A > D = H > B = C > E = F = G

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