Two mesh currents and flow in a network with a resistor carrying only

PHYSICS · Multi-part · 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 mesh currents I1I_1 and I2I_2 flow in a network with a resistor R1=8 ΩR_1 = 8\,\Omega carrying only I1I_1, a resistor R2=5 ΩR_2 = 5\,\Omega carrying only I2I_2, and a resistor Rm=3 ΩR_m = 3\,\Omega shared by both meshes, carrying I1−I2I_1-I_2. Battery voltages V1=10 VV_1 = 10\,\mathrm{V} drives mesh 1 and V2=6 VV_2 = 6\,\mathrm{V} drives mesh 2. Kirchhoff's voltage law around each mesh gives (R1+Rm)I1−RmI2=V1(R_1+R_m)I_1 - R_m I_2 = V_1 and −RmI1+(R2+Rm)I2=V2-R_m I_1 + (R_2+R_m)I_2 = V_2. ((a)) Write these two equations as AI=V\mathsf{A}\mathbf{I} = \mathbf{V}, giving the numerical entries of A\mathsf{A}, and state what row 1 and column 1 of A\mathsf{A} mean physically. ((b)) Evaluate det⁡A\det\mathsf{A} and say whether the currents are uniquely determined. ((c)) Is A\mathsf{A} symmetric? Explain in one sentence, in terms of the shared resistor, why this is physically sensible.

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