Four resistors are connected to a battery as shown in Figure P21.40. The current in the battery is I, the battery emf is ε, and the resistor values are R1 = R, R2 = 2R, R3 = 4R, and R4 = 3R. (a) Rank the resistors according to the potential difference across them, from largest to smallest. Note any cases of equal potential differences. (b) Determine the potential difference across each resistor in terms of ε. (c) Rank the resistors according to the current in them, from largest to smallest. Note any cases of equal currents. (d) Determine the current in each resistor in terms of I. (e) If R3 is increased, what happens to the current in each of the resistors? (f) In the limit that R3 → ∞, what are the new values of the current in each resistor in terms of I, the original current in the battery?
Figure P21.40
(a)
The rank of the resistors according to the potential difference across them from largest to smallest.
Answer to Problem 40P
The rank the resistors according to the potential difference across them from largest to smallest is
Explanation of Solution
The resistor
Write the equivalent resistance for the combination of resistors
Here,
The
Write the resistance of the circuit.
Here,
In series combinations, the potential is dropped in proportion to the resistance.
Resistor
Conclusion:
Therefore, the rank of resistors according to the potential difference across them from largest to smallest is
(b)
The potential difference across each resistors in terms of
Answer to Problem 40P
The potential difference across
Explanation of Solution
In series combinations, the potential is dropped in proportion to the resistance.
Resistor
Conclusion:
Therefore, the potential difference across
(c)
The rank of the resistors according to the current in them from largest to smallest.
Answer to Problem 40P
The rank of the resistors according to the current in them from largest to smallest is
Explanation of Solution
All the current will pass through the resistor
The ranking of the resistors according to the current in them from largest to smallest is
Conclusion:
Therefore, the rank resistors according to the current in them from largest to smallest is
(d)
The current in each resistor in terms of
Answer to Problem 40P
The current in
Explanation of Solution
The resistor
As a result, the current in
Conclusion:
Therefore, the current in
(e)
If
Answer to Problem 40P
The value of current
Explanation of Solution
As the value of resistance
Conclusion:
Therefore, the value of current
(f)
If
Answer to Problem 40P
The current
Explanation of Solution
If the
Conclusion:
Therefore, the current
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Chapter 21 Solutions
Principles of Physics: A Calculus-Based Text
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