Two cells, of and , each with internal resistance , are joined in parallel, positive…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Matrices, determinants and systems of linear equations

Problem

Two cells, of

6.0\,\mathrm{V}

and

5.0\,\mathrm{V}

, each with internal resistance

r

, are joined in parallel, positive terminal to positive terminal, across a load

R = 10\,\Omega

. Let

I_a

and

I_b

be the currents driven out of the two cells; they join in

R

. (a) Write the loop through each cell and the load as

\mathsf{A}\mathbf{I} = \mathbf{b}

, and show that

\det\mathsf{A} = r(2R + r)

. (b) For

r = 0.10\,\Omega

, find

I_a

I_b

and the current through the load. (c) With ideal cells,

r = 0

, the equations had no solution. Find what happens to the load current and to

I_a - I_b

r \to 0

, and say what the missing solution meant physically. (d) Now let both cells be

6.0\,\mathrm{V}

. At

r = 0

the equations have a whole family of solutions. Find

I_a

and

I_b

for

r > 0

, say which member of the family the real circuit approaches as

r \to 0

, and explain why the ideal equations could not pick it.

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