Question

The two 10-cm-long parallel wires in the figure are separated by 5.0 mm. For what value of the resistor R will the force between the two wires be 1.26×10−4 N?

Answer 1

To calculate the value of the resistor R, we can use the equation for force between two parallel wires. By equating the calculated force per unit length with the given force per unit length, we can solve for R. The value of resistor R in this case is 100 Ω.

Step-by-step explanation:

To calculate the force between two parallel wires, we can use the equation:

force per unit length = (μ₀ × current₁ × current₂)/(2π × distance)

Given that the wires are 10 cm long and separated by 5.0 mm, we have:

force per unit length = (4π × 10⁻⁷ T · m/A) × (5 × 10⁻³ A) × (5 × 10⁻² m) / (2π × 0.005 m)

Simplifying the expression, we get:

force per unit length = 125 × 10⁻⁷ N/m

Since the force between the wires is given as 1.26 × 10⁻⁴ N, we can set up an equation:

force per unit length = (1.26 × 10⁻⁴ N) / (10 cm)

Solving for force per unit length, we find:

force per unit length = 12.6 × 10⁻⁶ N/m

Equating the two expressions for force per unit length, we get:

(125 × 10⁻⁷ N/m) = (12.6 × 10⁻⁶ N/m)

Algebraically solving for the unknown resistor, R:

R = (12.6 × 10⁻⁶ N/m) / (125 × 10⁻⁷ N/m)

R = 100 Ω

Answer 2

To calculate the force between the two parallel wires, we can use the formula: F = µ₀ × I₁ × I₂ × (ℓ / (2πd)). Substituting the given values, the force is found to be F = 20 x 10⁽⁻⁶⁾ N. Therefore, the resistor R would need to have a value of 6.3 for the force between the wires to be 1.26 x 10⁽⁻⁴⁾ N.

Step-by-step explanation:

To calculate the force between the two parallel wires, we can use the formula:

F = µ₀ × I₁ × I₂ × (ℓ / (2πd))

Where:

• F is the force between the wires
• µ₀ is the permeability of free space (4π x 10⁽⁻⁷⁾ T · m/A)
• I₁ and I₂ are the currents in the wires (in this case, both are 5.0 A)
• ℓ is the length of the wires (10 cm or 0.1 m)
• d is the distance between the wires (5.0 mm or 0.005 m)

Now, we can substitute the values into the formula:

F = (4π x 10⁽⁻⁷⁾ T · m/A) x (5.0 A) x (5.0 A) x (0.1 m / (2π x 0.005 m))

F = (4π x 10⁽⁻⁷⁾ T · m/A) x 25.0 A² x (0.1 m / (2π x 0.005 m))

F = (4π x 10⁷⁾ T · m/A) x 25.0 A² x 10/(2π)

F = 20 × 10⁽⁻⁶⁾ N

So, in order for the force between the wires to be 1.26 x 10⁽⁻⁴⁾ N, the value of the resistor R would need to be 1.26 x 10⁽⁻⁴⁾ N / 20 x 10⁽⁻⁶⁾N = 6.3.