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Circuit worksheet—Capacitors Physics 104 We can treat circuits with capacitors in similar way to the circuits with resistors. We still find that voltages across parallel elements are the same. Now instead of having currents in series elements be the same, charges are the same. We can use a similar methodology where we reduce the sets of capacitors to an equivalent capacitance and use the ideas of series charge being the same and parallel voltages being the same. We can use the following set of equations. C1 In this circuit, C1 =2.0µF, C2=1.0µF, C3=3.0µF, and V=20V. Q = CV 1 Cseries = 1 1 1 + + C1 C2 C3 V C2 C3 C|| = C1 + C2 + C3 Begin by reducing C2 and C3 to an equivalent capacitance and redraw the circuit. C1 Resolve the parallel set: C23 = C2 + C3 C23 = 1.0 + 3.0 =4.0 μF C23 V Next reduce the remaining capacitors to one and redraw the circuit. Remaining capacitors are in series: C123 1 1 1 = + C123 C1 C23 2)( 4 ) ( C1C23 = = = 1.33 μF C1 + C23 (2) + ( 4 ) V Now use the idea that series capacitors have the same charge and parallel capacitors have the same voltage to find the charge, Q, and the voltage, V, on each of the capacitors. Charges on C1 and C23 are equal and equal to that on C123 V− Q1 = Q23 = Q123 ⎛1 Q Q Q1 Q23 1 ⎞ − = V − 123 + 123 = V − Q123 ⎜⎜ + ⎟⎟ = 0 C1 C23 C1 C23 ⎝ C1 C23 ⎠ Q1 = Q23 = VC123 = (20V )(1.33μF ) = 26.6 μC C123 Circuit worksheet—Capacitors Physics 104 Q123 Q123 V1 = & V23 = C1 C23 Voltages across C1 and C23 are not equal and sum to V Q123 26.6 μC = = 13.3 V C1 2.0 μF Q 26.6 μC = 6.65 V V23 = 123 = C23 4.0 μF V1 = Voltages across C2 and C3 are equal and equal to that on C23 Charges on C1 and C23 are not equal and sum to Q23 V1 + V23 = V V23 = V2 = V3 = 6.65 V Q123 = Q2 = V23C2 = ( 6.65 V )(1 μF ) = 6.65 μC Q3 = V23C3 = ( 6.65 V )(3 μF ) = 19.95 μC V23 = Q2 + Q3 C23 Q2 + Q3 = Q23 Summarize the capacitance, charge and voltage of each capacitor C1 = 2.0 μF Q1 = 26.6 μC V1 = 13.3 V C1 V C2 = 1.0 μF Q2 = 6.65 μC V2 = 6.65 V C2 C3 Since C2 = 1.0 = Q/V, Q2 = V2 x 10-6 C3 = 3.0 μF Q3 = 19.95 μC V3 = 6.65 V