Solving Resistor Circuits: Videos & Practice Problems

To solve complex circuit problems, understand that resistors can be in series or parallel. In series, current remains constant across resistors, while in parallel, voltage is constant. Use Ohm's law, V = I⁢R, to find unknown values. Start by simplifying the circuit to one equivalent resistor, then calculate voltage and current for each resistor, ensuring to apply Kirchhoff's laws for current conservation. This systematic approach helps in accurately determining circuit parameters.

Solving Resistor Circuits

In solving complex circuit problems, it's essential to understand how resistors behave in series and parallel configurations. When resistors are connected in series, the total or equivalent resistance can be calculated by simply adding their resistances together. The formula for equivalent resistance in series is:

R_{eq} = R_1 + R_2 + R_3 + ... + R_n

Conversely, for resistors in parallel, the equivalent resistance is found using the reciprocal formula:

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}

In series circuits, the current remains constant across all resistors. For example, if a series circuit has a current of 2 amps, each resistor in that series will also carry 2 amps. This is crucial when determining the current through an equivalent resistor formed by combining series resistors.

In parallel circuits, the voltage across each resistor is the same. If three 9-ohm resistors are connected in parallel and the voltage across them is 5 volts, then each resistor experiences 5 volts. The equivalent resistance for these resistors can be quickly calculated using the shortcut for three identical resistors:

R_{eq} = \frac{R}{n} = \frac{9 \, \text{ohms}}{3} = 3 \, \text{ohms}

To solve circuit problems systematically, follow these three steps:

1. Reduce the circuit to a single equivalent resistor using series and parallel combinations.
2. Apply Ohm's Law, V = IR, to find the voltage and current for the equivalent resistor.
3. Work backwards to determine the voltage and current for each individual resistor based on the relationships established in the first two steps.

For instance, if you have a circuit with a 10-volt battery and a combination of resistors, first find the equivalent resistance. Then, using Ohm's Law, calculate the current drawn from the battery. This current will flow through all series resistors, allowing you to find their individual voltages using the same law.

When working backwards, remember that resistors in series share the same current, while resistors in parallel share the same voltage. This understanding allows you to calculate the unknown values for each resistor effectively.

Finally, ensure that your calculations adhere to Kirchhoff's Current Law, which states that the total current entering a junction must equal the total current leaving it. This principle can help verify your results and ensure consistency throughout your calculations.

By maintaining an organized approach and systematically applying these principles, you can effectively tackle complex circuit problems and gain a deeper understanding of electrical circuits.

What is current and voltage across each resistor below?

• A. V_2Ω=5.0 V, i_2Ω=4.44 A;
  V_1Ω=5.0 V, i_1Ω=2.22 A;
  V_3Ω=5.0 V, i_3Ω=6.67 A
• B. V_2Ω=6.67 V, i_2Ω=0.500 A;
  V_1Ω=3.33 V, i_1Ω=0.500 A;
  V_3Ω=10.0 V, i_3Ω=2.23 A
• C. V_2Ω= 1.82V, i_2Ω=0.908 A;
  V_1Ω=1.82 V, i_1Ω=0.908 A;
  V_3Ω=8.18 V, i_3Ω=1.36 A
• D. V_2Ω=6.67 V, i_2Ω=3.33 A;
  V_1Ω=3.33 V, i_1Ω=3.33 A;
  V_3Ω=10.0 V, i_3Ω=3.33 A

Answer: (D)

Find Current of One Resistor

In this example, we explore how to determine the current flowing through a 3-ohm resistor in a parallel circuit. The key to solving this problem lies in understanding Ohm's Law, which states that voltage (V) equals current (I) multiplied by resistance (R). This relationship can be rearranged to find current as follows:

\[I = \frac{V}{R}\]

In this case, we know the resistance (R) is 3 ohms. To find the current, we need the voltage (V) across the resistor. A crucial property of parallel circuits is that the voltage across all components connected in parallel is the same. Therefore, if the voltage supplied by the battery is 5 volts, the voltage across the 3-ohm resistor is also 5 volts.

Substituting the known values into the equation gives us:

\[I = \frac{5 \text{ volts}}{3 \text{ ohms}} = 1.67 \text{ amps}\]

This straightforward calculation demonstrates the importance of recognizing circuit properties, such as the equality of voltage in parallel configurations. By applying these principles, even seemingly complex problems can be simplified to manageable calculations. Understanding these concepts prepares you for a variety of questions, whether they appear simple or complex.

What is the voltage of the battery below?

• A. 1.71 V
• B. 5.25 V
• C. 11.3 V
• D. 13.0 V

Answer: (C)

Here’s what students ask on this topic:

How do you find the equivalent resistance in a series circuit?

To find the equivalent resistance in a series circuit, simply add the resistances of all the resistors together. The formula is:

$R_{\mathrm{eq}}=R_1+R_2+R_3+...+R_n$

where $R_{\mathrm{eq}}$ is the equivalent resistance, and $R_1$, $R_2$, $R_3$, ..., $R_n$ are the resistances of the individual resistors.

How do you calculate the equivalent resistance in a parallel circuit?

To calculate the equivalent resistance in a parallel circuit, use the reciprocal formula:

$R_{\mathrm{eq}}=\frac{1}{\frac{1}{R_1}}$

where $R_{\mathrm{eq}}$ is the equivalent resistance, and $R_1$, $R_2$, $R_3$, ..., $R_n$ are the resistances of the individual resistors.

What is Ohm's Law and how is it used in solving resistor circuits?

Ohm's Law states that the voltage (V) across a resistor is equal to the current (I) flowing through it multiplied by its resistance (R). The formula is:

$V=IR$

In solving resistor circuits, Ohm's Law is used to find unknown values of voltage, current, or resistance. For example, if you know the current and resistance, you can find the voltage. Similarly, if you know the voltage and resistance, you can find the current.

How do you apply Kirchhoff's laws to solve complex resistor circuits?

Kirchhoff's laws include the Junction Rule and the Loop Rule. The Junction Rule states that the total current entering a junction equals the total current leaving the junction. The Loop Rule states that the sum of the voltage gains and drops around any closed loop in a circuit must be zero. To solve complex resistor circuits, use these laws to set up equations based on the currents and voltages in the circuit. Solve these equations simultaneously to find the unknown values.

What are the steps to solve a complex resistor network?

To solve a complex resistor network, follow these steps:

1. Simplify the circuit to one equivalent resistor by combining resistors in series and parallel.
2. Calculate the voltage and current for the equivalent resistor using Ohm's Law.
3. Work backwards to find the voltage and current for each individual resistor, applying Kirchhoff's laws as needed.

This systematic approach ensures accurate determination of circuit parameters.