Ch 29: The Magnetic Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 29: The Magnetic Field

Problem 33

Chapter 29, Problem 33

The two 10-cm-long parallel wires in FIGURE EX29.33 are separated by 5.0 mm. For what value of the resistor R will the force between the two wires be 5.4 x 10⁻⁵ N?

Verified step by step guidance

Step 1: Understand the problem. The force between two parallel current-carrying wires is given by the formula:

F = rac{{\(\text{μ₀}\) 	imes I₁ 	imes I₂ 	imes L}}{{2\(\text{π}\) 	imes d}}

, where F is the force, μ₀ is the permeability of free space (

4\(\text{π}\) 	imes 10^{-7} \(\text{ T·m/A}\)

), I₁ and I₂ are the currents in the wires, L is the length of the wires, and d is the distance between the wires.

Step 2: Rearrange the formula to solve for the product of currents I₁ × I₂. Using the given values, substitute F = 5.4 × 10⁻⁵ N, L = 0.10 m, and d = 0.005 m into the formula:

I₁ 	imes I₂ = rac{{F 	imes 2\(\text{π}\) 	imes d}}{{\(\text{μ₀}\) 	imes L}}

Step 3: Recognize that the current I₁ in one wire is determined by the circuit containing the resistor R. Use Ohm's Law,

I₁ = rac{{V}}{{R}}

, where V is the voltage across the resistor R. Substitute this expression for I₁ into the formula for I₁ × I₂.

Step 4: Assume the second wire carries the same current as the first wire (I₂ = I₁). Substitute I₂ = I₁ into the formula for I₁ × I₂, resulting in

I₁^2 = rac{{F 	imes 2\(\text{π}\) 	imes d}}{{\(\text{μ₀}\) 	imes L}}

. Solve for I₁ by taking the square root:

I₁ = \(\text{√}\)rac{{F 	imes 2\(\text{π}\) 	imes d}}{{\(\text{μ₀}\) 	imes L}}

Step 5: Substitute the expression for I₁ from Step 4 into Ohm's Law to find R:

R = rac{{V}}{{I₁}}

. Use the calculated value of I₁ and the given voltage V to determine the resistor value R.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Force Between Parallel Wires

The magnetic force between two parallel wires carrying current is given by the formula F = (μ₀/2π) * (I₁ * I₂ * L) / d, where F is the force, μ₀ is the permeability of free space, I₁ and I₂ are the currents in the wires, L is the length of the wires, and d is the distance between them. This relationship shows how the force depends on the currents and their separation.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. It is mathematically expressed as V = I * R, which is essential for determining the current in the wires when a voltage is applied.

Resistor in a Circuit

A resistor is a component in an electrical circuit that resists the flow of electric current, causing a voltage drop. The value of the resistor affects the current flowing through the circuit according to Ohm's Law. In this problem, finding the appropriate resistor value R is crucial to achieve the desired force between the wires by controlling the current.

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