Answers on separate paper. All answers must be justified by calculations and/or explanation. Draw diagrams where appropriate. Partial credit is available.
1) The circuit below consists of a group of external resistors and a battery with a 9V EMF which has a 1 Ω internal resistance. Determine
a) The current in each resistor. It should be clear (when I read your paper) as to which current goes with each resistor.
b) The terminal voltage of the battery,
c) The total power input (rate at which chemical energy is consumed) using εI, and the power output for each resistor individually, including the internal resistance, by using I2R (I is the current in the individual resistor). Sum these individual outputs and show that the power output is equal to the power input.
2) Since the terminal Voltage of a battery is given by ΔV = ε - Ir where ε is the emf, I the current in the battery and r the internal resistance, the power output of the battery is given by P = IΔV = εI - I2r . P is then the power transferred to the external circuit.
a) Since the power output is a quadratic function of the current, there is a value of I for which the power reaches a maximum. Use calculus and determine both the value of the current for which there is maximum power and the maximum amount of power at this current in terms of the givens ε and r. You have to clearly show your work.
b) At what rate is thermal energy developed in the battery when it transfers maximum power to the circuit )? At what rate is chemical energy consumed in the battery at maximum power? (Answer both in terms of the givens ε and r
c) When the maximum amount of current that can be drawn from the battery is drawn the terminal voltage will be zero. In terms of the givens ε and r how much current is this and how much power is transferred to the circuit under these conditions?
3) In the circuit below a battery with a 9 V emf and a 2 Ω internal resistance is connected to two resistors and a switch that connects points a and b. The + and - represent the poles of the battery.
a) With the switch open determine the current in the battery and the current in each resistor.
b) Determine the terminal voltage of the battery.
c) Now the switch is closed. To make the math easier assume that the resistance between points a and b in the circuit is small but finite: Rab=0.1 Ω. Determine the terminal voltage across the battery and the current in the 20 Ω resistor.
d) Compare the power output of the 20 Ω resistor with the switch open to the output when the switch is closed. If this resistor were a light bulb how would closing the switch affect its behavior?
4) In the circuit below there are three batteries each with the same internal resistance r and three external resistors, each with resistance R. r=0.4Ω and R=5Ω.
a) Redraw the circuit diagram on your solution sheet and label the unknown currents in the circuit. Apply Kirchoff's rules to the circuit to solve for the unknown currents. One of these equations must be the junction rule applied to the circuit. The remaining equations are applications of the loop rule: show the equations, and label the loops on the diagram. Solve the equations for the unknown currents.
b) If the external resistors are light bulbs, which (if any) is the brightest bulb based on your results in part a? Justify your answer. At what rate is thermal energy developed in this bulb? In which battery is the heating (thermal energy) rate the greatest? (Again based on your results and justify your answer.)
c) Use your results to determine the terminal voltage of all three batteries. Are any of the batteries charging? How can you tell?
Note: We will discuss matrix based methods in class to solve systems of linear equations such as those generated for this circuit. You may use your calculators to implement these methods.