Question

A resistor of unknown resistance and a \(35-\Omega\) resistor are connected across a \(120-\mathrm{V}\) emf device in such a way that an 11 -A current flows. What is the value of the unknown resistance?

Short answer

Answer: The value of the unknown resistor is approximately 16 Ω.

Step-by-step solution

Determine the type of connection

Since the total current flowing through the circuit is given, this implies that the resistors are connected in parallel. In a parallel connection, the total current is the sum of the currents flowing through each branch of the resistors.

Apply Ohm's Law to the known resistor

Ohm's Law states that the voltage across a resistor (\(V_R\)) is equal to the product of the current through the resistor (\(I_R\)) and the resistor's resistance (\(R\)). In mathematical form: \(V_R = I_RR\). We apply Ohm's Law to the 35-Ω resistor to find its current. Let's call this current \(I_1\). \(120 \mathrm{V} = I_1 (35 \Omega)\) Solving for \(I_1\), we get: \(I_1 = \dfrac{120 \mathrm{V}}{35 \Omega} = 3.43 \,\text{A}\)

Find the current through the unknown resistor

Since the total current in the circuit is 11-A, we can find the current through the unknown resistor by subtracting the current through the 35-Ω resistor from the total current. Let's call the current through the unknown resistor \(I_2\). \(I_2 = 11 \,\text{A} - 3.43 \,\text{A} = 7.57 \,\text{A}\)

Apply Ohm's Law again to find the unknown resistance

Now, we apply Ohm's Law to the unknown resistor, using the calculated current \(I_2\) and the given voltage of the emf device. Let's call the unknown resistance \(R_2\). \(120 \mathrm{V} = 7.57 \,\text{A}R_2\) Solving for \(R_2\), we get: \(R_2= \dfrac{120\mathrm{V}}{7.57 \,\text{A}} = 15.85 \,\Omega\)

Round off the answer to an appropriate number of significant figures

Given values in this exercise have two significant figures; hence the result should be rounded off to two significant figures. So, the value of the unknown resistance is approximately \(16\,\Omega\).

Key concepts

Resistor

A resistor is an essential component in electrical circuits, widely used to control the flow of electric current. It transforms electrical energy into heat, providing resistance measured in ohms (\(\Omega\)).
Resistors can be found in various applications from simple household electronics to complex industrial systems.

• They are crucial in protecting sensitive components by ensuring the current doesn't exceed safe levels.
• Resistors limit current to desirable levels, facilitating multiple functions within a circuit.

Resistors maintain their importance by contributing to functions such as voltage division, biasing active elements, and in circuitry for sensors and transducers.

Parallel Circuit

Parallel circuits allow multiple components to share the same voltage source, distributing current across various branches. They are characterized by having a clear path for current to flow through each resistor.
In a parallel circuit, the total voltage remains the same across each branch, and the total current is the sum of the currents flowing through each resistor.

• This means each resistor receives the full voltage of the source.
• Adding more resistors in parallel decreases the total resistance.

The beauty of parallel circuits lies in their ability to keep other components unaffected if one path fails, thereby maintaining operation in everyday devices like home lighting systems and appliances.

Current Flow

Current flow refers to the movement of electric charge through a conductor in a circuit. It is typically measured in amperes (A), and crucially determines how components like resistors influence the circuit performance.
Understanding current flow helps in calculating how much electricity moves through different parts of a circuit.

• Ohm’s Law, expressed as \(V = IR\), is used to relate current, voltage, and resistance in circuits.
• In parallel circuits, total current is the sum of currents through each branch, as witnessed in the given exercise.

The explanation of Ohm’s Law helps students grasp how resistors affect the flow of current, a fundamental concept for designing and analyzing electrical circuits.