Ch 29: The Magnetic Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 29: The Magnetic Field

문제 33

29장, 문제 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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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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