Solving Resistor Circuits: Videos & Practice Problems

An experimental circuit is shown below. All meters and batteries are ideal. When the 18.0 Ω resistor is plucked out (creating a gap), the voltmeter displays 12.0 V. Determine the ammeter reading when the 18.0 Ω is replaced with a low resistance shunt. 

Determine the discharge time constant of the capacitors shown in the figure provided.

How much time does it take for the charge on the capacitor in an RC circuit with a time constant of 20 ms to i) decrease to one-fourth of its initial value? ii) Additionally, after the discharge begins, how long does it take for the energy stored in the capacitor to decrease to 75% of its initial value?

Consider a 200 μF capacitor charged to an initial voltage of 1800 V. If it loses 85% of its charge within 25 ms when discharged through a certain component, determine the resistance of this component.

In a specific circuit arrangement, a 25 μF capacitor initially charged to 10 V is linked in parallel with a resistor. As the capacitor gradually discharges, its stored energy decreases by 60% within a time span of 0.3 s. Determine the resistance value required for this circuit.

Consider the following circuit configuration. Find an expression for the current i, as a function of time 't'.

As a part of the competition, you are provided with 2.0 g of copper and tasked with creating a wire that can dissipate 10.0 W of power when connected to a 2.0 V power source. Calculate the length and diameter of the copper wire to meet this requirement. The resistivity of copper is 1.7 × 10^-8 Ω-m and the density of copper is 8.94 × 10³ kg/m³.

A pair of metal pieces, both measuring 8 cm × 8 cm, are positioned 0.5 cm apart. One of the pieces has a +15 nC charge while the other is charged to -7.5 nC. The two metal pieces are connected to each other by a metallic string. Will the electric current through the circuit exhibit a tendency to increase, decrease, or remain constant over time? 

Following the initial charging of a 200 μF capacitor to 12 V, it is connected in parallel with a 3000 Ω resistor. The capacitor discharges through the resistor, although at a slower rate compared to a direct wire connection of the plates. If the fully charged capacitor is connected to the resistor at t = 0 s, determine the time at which the capacitor voltage decreases by a quarter to reach 9.0 V.

A 3-A current passes through a cylindrical conductor with a radius of 4 mm. The current density in the conductor, which varies along the radial direction, is defined by J = Jₑ(r/R)², where Jₑ is the current density at its surface. What is the value of Jₑ?

Determine the current that would flow through a 3.0-meter-long wire with a radius of 0.25 mm, which has a resistivity described by the equation ρ(x) = (3.0 × 10^-7 Ω•m) + (4.0 × 10^-7 Ω)x, when it is connected to a 12.0 V battery.

Consider an electric circuit where the amount of charge flowing through a wire over time t is given by the function Q(t) = 2sin(t) + 3t, where t represents time in seconds. Express the current in the wire as a function of time.

The amount of charge flowing through a conductor over time is given by the function Q(t) = 5t  ² - 3t, where t is in seconds, and Q cannot exceed a maximum value of 10 coulombs. Plot a graph of the current (I) versus time (t) over the interval t in which the amount of Q is allowed.

A capacitor is being charged through a resistor in an electric circuit. The total charge that has entered the capacitor at time t is given by the equation Q = (10 μC) (1 - e^{(-t/5 s)}), where t is the time in seconds. What is the maximum value of the current flowing through the resistor?

A capacitor is being discharged through a resistor in an electric circuit. The current flowing through the resistor decreases over time as the capacitor discharges. The current as a function of time is given by the equation I = (2 A)e^{-t/5 s}, where t represents time in seconds since the discharging process started. Determine the total charge that has flowed out of the capacitor from the initial time until the current becomes zero.

Determine the unknown resistance R_a in the electric circuit shown below. 

Consider the electronic circuit shown in the figure below. The circuit comprises two unknown resistors, denoted as R_x, and a third resistor with a known value of 50 Ω. The equivalent resistance between points A and B is measured to be 18.75 Ω. Determine the value of R_x.

A capacitor of 15 μF capacitor is discharged through an unknown resistor. The current flowing through the resistor is measured using an ammeter. The recorded data is presented in the table below. Determine:

i) the resistance

ii) the initial voltage across the capacitor.

A 1.5-V AA battery is powering a small LED light with a resistance of 5 Ω. When the LED light is on, the voltage across the battery terminals is measured to be 1.3 V. Calculate the internal resistance of the battery.

To minimize power dissipation in the circuit shown, which specific pair of points should a 9 V battery be connected to?

A solar-powered device uses two 12.0 V batteries connected in series. The resistance of the device is 22.0 Ω. The power supplied to the device decreases as the battery power gets depleted due to increased internal resistance. Assuming zero initial resistance, determine the total internal resistance from both batteries when the power supplied by the batteries to the device is a quarter of the initial value.

In the circuit below, find the current through the 4-Ω resistor.

Imagine working in a physics lab with a galvanometer with a sensitivity of 30 kΩ/V and an internal resistance of 15 Ω. Your task is to modify this galvanometer to function as an ammeter capable of reading up to 2.0 A at full scale. How would you achieve the desired output?

You have a galvanometer with an internal resistance of 40 Ω that shows full-scale deflection with a 60 μA current. You need to modify this galvanometer to function as a voltmeter that measures up to 300 V. How can you achieve the desired results?

You are in a physics lab working with a milliammeter with a full-scale reading of 50 mA. It contains a 0.10-ohm resistor in parallel with a 50-ohm galvanometer. You now need to use this instrument to measure voltage, with a full-scale reading of 50 V, and you cannot disassemble the ammeter. How would you modify it to serve as a voltmeter, and what would be its sensitivity in ohms per volt?

A strain gauge sensor is affixed to a metal rod to measure small changes in its length. The sensor has an initial length of L₀, a cross-sectional area of A₀, and a resistance of R₀. As the rod deforms, the sensor's length changes by ΔL, and the resistance changes by ΔR. Assuming that the volume of the sensor remains constant, derive an expression approximating the strain: $\frac{\Delta L}{L_0}$ , in terms of ΔR and R₀.

A dielectric material with resistivity of "ρ" and dielectric constant of "K" is placed between the plates of a parallel-plate capacitor. The capacitor, which is disconnected from a battery, has charges of +Q and -Q on these plates. Modeling the system with the dielectric as a resistor in parallel with the capacitor, determine the time constant "τ" for discharge if the dielectric is rubber with 𝜌 = 2.0×10¹³ Ω⋅m and K = 3.0.

A 2.0 μF capacitor is to be charged to 2500 V in 3.0 seconds. Assume that this 2500 V is 90% of the full source voltage. Determine the resistance that is required.

If four 30 Ω resistors and two 50 Ω$\Omega$ resistors are connected in series, find the circuit's total resistance, ignoring the battery's internal resistance.

Five 20 Ω resistors and two 40 Ω resistors are connected in parallel. What is the total resistance of the circuit?

If two LEDs are connected to a 200 V DC source in series, they will consume a quarter of the power source compared to the condition when they are connected in parallel instead. Given that the resistance of one of the LEDs is 3.9 kΩ, calculate the resistance of the other LED. Assume that the internal resistance of the DC source is zero.

Two lightbulbs with resistances 1.8-kΩ and 2.2-kΩ are connected in parallel. This parallel combination is then connected with a 1.6-kΩ lightbulb in series. Given that each bulb can produce a maximum of 0.8 W before it gets fused, calculate the maximum voltage that can be applied across the entire network. Ignore the internal resistance of the DC source.

Two light bulbs are connected in series with a 12.5-V power supply. The first bulb has a resistance of 10 kΩ. A voltmeter with 18.5 kΩ internal resistance measures 4.2 V across the second bulb. What is the resistance of the second bulb?

At 2.50 kΩ each, three resistors sit on your desk, and you’re curious about how connecting them in different ways might change the overall resistance in a circuit. You wonder, what configurations you can make with these resistors, and how the net resistance will differ for each case. First, figure out the different configurations you can create and then calculate the net resistance for each one.

A 12.0 m cable consists of 6.0 m of Fe followed by 6.0 m of Ni, both with a diameter of 1.6 mm. Across the entire length of the cable, a potential difference of 120 mV is applied. What is i. the total resistance of the cable, ii. the current flowing through the cable, and iii. the voltage across the Ni and Fe sections of the cable? Hint: Use the following resistivities: Ni = 6.84 × 10 ^-8 Ω·m, Fe = 9.71 × 10^-8 Ω·m.

During an experiment, a biophysicist connects a charged capacitor across a frog leg with a resistance of 1000 Ω. The capacitor exhibits a time constant of 15 ms as it discharges. Determine the effective capacitance of the frog leg.

A physics lab experiment involves measuring the potential difference across a 30 MΩ component in a high-voltage test circuit. The lab's voltmeter has an internal resistance of 20 MΩ and is rated for a maximum of 300 V. To safely measure voltages up to 30 kV, a resistor is to be placed in series with the 30 MΩ component. What should be the resistance value of this resistor if the voltmeter is to read 150 V when the circuit is at 30 kV?

John is testing an "unknown" power supply using a 5.0-kΩ resistor. He measures a current of 3.65 mA flowing through the resistor. What is the terminal voltage 𝑉_𝑎𝑏 of the power supply? (There are two possible answers. Why might this be the case?).

A physics student designs a circuit in the laboratory and connects it with a voltage source as shown in the figure below. Calculate the time constant of the circuit for charging the capacitors after the 12 V voltage source is applied.

In the figure below, R₁ = 20 Ω, R₂ = 15 Ω, and R₃ = 10 Ω. What is the equivalent resistance of the circuit? (Assume that the internal resistance of the battery is zero.)

An electrical engineering student is designing a circuit that includes three resistors of 2.0 Ω, 4.0 Ω, and 8.0 Ω, connected as shown in the figure. Assuming that the internal resistance of the battery is zero, calculate the potential drops across the resistors V₁, V₂, and V₃ as indicated in the figure.

A circuit is designed with two resistors in parallel, having resistances of 10 Ω and 15 Ω. This parallel combination is then connected in series with a third resistor that has a resistance of 5.0 Ω. This configuration of resistors is connected across a 22 V battery. Given that the internal resistance of the battery is negligible, calculate the currents through each of the resistors.

A tank filled with two layers—cold water at the bottom (temperature ≤ 4°C) and warm air above it (temperature > 10°C)—has a temperature-sensitive metal rod passing vertically through it. The length of the rod, L, spans the height of the tank. The rod, made of a special alloy, exhibits a superconductor below 10°C and behaves as a normal conductor with resistivity ρ above this temperature. A voltmeter measures the voltage V across the rod while a constant electrical current I flows through it. The section of the rod exposed to warm air exhibits normal resistance, while the section submerged in cold water has very low resistance. Let g = y/L, where y is the height of the rod submerged in water, representing the fraction of the rod in cold water. Let V₀ be the voltage across the rod when it is completely exposed to warm air (i.e., g = 0). Derive the expression for g in terms of V and V₀.

In a laboratory setup, a heating system is used to maintain a chemical solution at a constant temperature. The heating element has a resistance R_heat that may fluctuate slightly around a nominal value R_heat,0 by a small amount ΔR_heat, where ΔR_heat << R_heat,0. To ensure stable power delivery to the heating element, a circuit is designed as shown in the figure below. The circuit includes two stabilizing resistors, each with resistance R_stab, connected in series with each other and in parallel with the heating element R_heat.

Determine the value of R_stab that will keep the power delivered to the heating element approximately constant, even if R_heat fluctuates slightly.

Hint: For small changes in resistance, $\left.\Delta P \approx \Delta R_{\text {heat }} \frac{d P}{d R_{\text {heat }}}\right|_{R_{\text {heat }}=R_{\text {heat }, 0}}$.

An electrical heating system in a building is powered by three resistive heating elements, each connected to separate voltage sources. The voltage sources, labeled V₁ and V₂, supply 120 V and 240 V, respectively. With resistances of R₁ = 10 Ω, R₂ = 15 Ω, and R₃ = 20 Ω, the heating elements form a mixed series-parallel circuit with the voltage sources. Determine the magnitudes and directions of the currents I₁, I₂, and I₃ flowing through each heating element, ignoring any internal resistance of the voltage sources.

A remote weather station is powered by two backup batteries with different voltage ratings to ensure reliability. The batteries have EMFs of 2.2 V and 3.5 V, and each has an internal resistance of r=0.400Ω. The batteries are connected in a circuit that powers a critical sensor through a resistor R=4.50 Ω. Determine the voltage across the resistor R that powers the sensor.

In the circuit shown below, three batteries and several resistors are connected in a complex network, with each battery having an internal resistance of r = 1.0 Ω. The currents through the branches of the circuit have been determined as I₁ = 0.511 A in the upper branch and I₂ = 0.508 A in the middle branch. Using these given current values, calculate the terminal voltage of the 6.0 V battery in the bottom branch of the circuit.

A 3-bit digital-to-analog converter (DAC) circuit converts binary inputs into analog voltages using switches and resistors. Each binary bit (2ⁿ) is linked to a switch that, if "1," connects the corresponding resistor to the circuit, while "0" leaves it disconnected. The circuit has an input voltage of 10 V and includes resistors of 10 kΩ, 5.0 kΩ, and 2.5 kΩ for bits 0, 1, and 2, respectively. A 2.0 Ω resistor is positioned at the output to measure the resulting analog voltage. Calculate the voltage across the 2.0 Ω resistor for the binary inputs 001, 010, 011, and 101 (which represent decimal values 1, 2, 3, and 5, respectively).

A physics student designs a circuit (as shown below). If the student correctly measures the potential difference between points 𝑝 and 𝑚, what value should they obtain?

A designer is working on a lighting project that requires a specific LED to operate at 4.50 V. The power source available is 9.00 V. If the designer decides to use a voltage divider with a resistor R of 10.0 Ω, (i) what should the resistance of the LED, R' be to achieve the desired voltage across the LED? (ii) If another LED of 15.0 Ω is connected in parallel to this LED (R'), what would the voltage across R' be then? (Assume that the internal resistance of the power source is zero.)

In a laboratory, a student connects seven 2.0 Ω resistors across a power source, as shown in the figure below. Calculate the equivalent resistance across the terminals P and Q. (Assume that the internal resistance of the power source is negligible.)

A physics student is studying series and parallel combinations in his laboratory with two resistors of 5.0 Ω and 15 Ω. He connects them in series and parallel and keeps the low voltage end at 0 V and the high voltage end at 4.0 V in each case. Afterward, he measures the currents through the resistors correctly in both cases. What should his measurements be? 

A scientist is testing a high-resistance circuit powered by a 20.000 V battery, containing a known 12.00 MΩ resistor (R₁₂) in series with an unknown resistor R. Using a sensitive voltmeter with internal resistance R_V, the scientist takes two measurements: first, when the voltmeter is connected across the 12.00 MΩ resistor, it reads V₁ = 512 mV; then, when the voltmeter is connected across the unknown resistor R, it reads V₂ = 6.824 V. Determine the value of the unknown resistor R based on these measurements.