A planar circuit contains two simple loops that share one resistor

PHYSICS · Subjective · AP · Physics 2: Algebra-based · Electric Circuits · Electric Circuits

Problem

A planar circuit contains two simple loops that share one resistor. The elements are labelled as follows: • Left loop outer arm: resistor

R_1

in series with source

\mathcal E_1

(positive terminal at the top of the loop). • Right loop outer arm: resistor

R_2

in series with source

\mathcal E_2

(positive terminal at the top of the loop). • The top ends of

R_1

and

R_2

are joined by a resistor

R_3

. The bottom ends of the two sources are connected together. Let clockwise loop currents

I_1

and

I_2

circulate in the left and right loops, respectively. (a) Write the pair of simultaneous linear equations that link

I_1

and

I_2

in terms of

R_1, R_2, R_3, \mathcal E_1,

and

\mathcal E_2

. (b) Solve the equations algebraically to obtain explicit expressions for

I_1

and

I_2

. (c) Show that if

\mathcal E_1 = \mathcal E_2

and

R_1 = R_2

, the current through the resistor

R_3

is zero, regardless of the numerical value of

R_3