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Chapter 21: Problem 36

A circuit consists of three resistors, \(R_{1}

Short Answer

Expert verified
For current: tie (I is same for all). For potential difference, order is: \( V_1 < V_2 < V_3 \).

Step by step solution

01

Understand Series Circuit

In a series circuit, the current through each component, like a resistor, is the same. This means the current through resistor \(R_1\) is equal to the current through \(R_2\) and \(R_3\).
02

Identify Current through Resistors

Since all resistors are in series, the current through all resistors is the same: \( I_1 = I_2 = I_3 \). Thus, there is a tie for the current through resistors.
03

Apply Ohm's Law

Use Ohm's Law, \( V = IR \), to find the potential difference across each resistor. Here, \( V_1 = I \times R_1 \), \( V_2 = I \times R_2 \), and \( V_3 = I \times R_3 \).
04

Rank Potential Difference

Rank the potential differences based on the resistances: Because \( R_1 < R_2 < R_3 \), it follows that \( V_1 < V_2 < V_3 \). Therefore, the potential difference is smallest across \( R_1 \) and largest across \( R_3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Current in Series Circuits
In a series circuit, the current flows through each component consecutively, like water flowing through connected pipes. Because of this continuous path, the same amount of current flows through each resistor, no matter their different sizes or values. This is because there is only a single path for the current to take. When you are dealing with resistors connected in series, you can rest assured that the current through each resistor remains constant. In simpler terms:
  • The current in a series circuit does not change from one component to the next.
  • It doesn't matter the order of the resistors; the current will be the same through all, such as in our example with resistors \(R_1, R_2,\) and \(R_3 \).
This behavior makes it simpler when calculating the current in complex circuits as you only need to know the total flowing through one part, and it applies to all parts.
Ohm's Law
Ohm's Law is a fundamental principle used to calculate relationships between voltage, current, and resistance in electrical circuits. The equation is given by:
  • \( V = IR \)
Where \(V\) stands for the potential difference (voltage) across a component, \(I\) is the current flowing through it, and \(R\) is the resistance. It's like understanding the pressure, flow, and friction in a hose.
Using Ohm's Law, once we know any two of these values, the third one can be easily calculated. For a series circuit, this equation can help us find the voltage drop across each resistor:
  • If the total resistance is high, say with \(R_3\) being the largest, it will have the largest voltage drop according to \( V_3 = I \times R_3 \).
  • The same current flows through each resistor, so multiplying by larger resistances gives larger potential differences.
Hence, knowing the resistance is crucial to determine the voltage distribution along the circuit.
Potential Difference in Series Circuits
The potential difference (or voltage) in a series circuit is not distributed equally unless the resistors are identical. Instead, it depends directly on the resistance each component presents.
Here's how the concept works:
  • Higher resistance translates to higher potential difference across that component.
  • If you were to walk through the circuit, starting fresh from the battery, the first resistor you'd hit, say \( R_1 \), if it has lower resistance, has less voltage drop compared to \( R_3 \) in our scenario where \( R_1 < R_2 < R_3 \).
Thus, ranking is straightforward based on resistance values:
  • \( V_1 < V_2 < V_3 \) such that higher resistance \( (R_3) \) leads to larger voltage drops compared to lower resistance \( (R_1) \).
By understanding this principle, you can effectively predict how different resistors affect the overall behavior of the circuit.

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Most popular questions from this chapter

Predict/Explain An electric space heater has a power rating of \(500 \mathrm{W}\) when connected to a given voltage \(V\). (a) If two of these heaters are connected in series to the same voltage, is the power consumed by the two heaters greater than, less than, or equal to 1000 W? (b) Choose the best explamation from among the following: I. Each heater consumes \(500 \mathrm{W}\); therefore two of them will consume \(500 \mathrm{W}+500 \mathrm{W}=1000 \mathrm{W}\) II. The voltage is the same, but the resistance is doubled by connecting the heaters in series. Therefore, the power consumed \(\left(P=V^{2} / R\right)\) is less than \(1000 \mathrm{W}\) III. Connecting two heaters in series doubles the resistance. since power depends on the resistance squared, it follows that the power consumed is greater than \(1000 \mathrm{W}\).

A 12-V battery is connected to three capacitors in series. The capacitors have the following capacitances: \(4.5 \mu \mathrm{F}, 12 \mu \mathrm{F},\) and \(32 \mu F .\) Find the voltage across the \(32-\mu F\) capacitor.

Predict/Explain Two capacitors are connected in parallel. (a) If a third capacitor is now connected in parallel with the original two, does the equivalent capacitance increase, decrease, or remain the same? (b) Choose the best explanafion from among the following: I. Adding a capacitor tends to increase the capacitance, but putting it in parallel tends to decrease the capacitance; therefore, the net result is no change. II. Adding a capacitor in parallel will increase the total amount of charge stored, and hence increase the equivalent capacitance. III. Adding a capacitor in parallel decreases the equivalent capacitance since each capacitor now has less voltage across it, and hence stores less charge.

Two resistors, \(R_{1}=R\) and \(R_{2}=2 R\), are connected to a battery, (a) Which resistor dissipates more power when they are connected to the battery in series? Explain. (b) Which resistor dissi- pates more power when they are connccted in parallel? Explain.

A current of 0.96 A flows through a copper wire \(0.44 \mathrm{mm}\) in diameter when it is connected to a potential difference of \(15 \mathrm{V}\). How long is the wire?
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