Question

Graph of force per unit length between two long parallel current carrying conductors and the distance between them (a) Straight line (b) Parabola (c) Ellipse (d) Rectangular hyperbola

Short answer

The force per unit length between two long parallel current-carrying conductors is given by f = I₂ × \(\frac{\mu_0 I_1}{2 \pi r}\), where f is the force per unit length, I₁ and I₂ are the currents in the conductors, and r is the distance between the conductors. Since f is inversely proportional to r, the graph shape is a rectangular hyperbola. Thus, the correct answer is (d) Rectangular hyperbola.

Step-by-step solution

Determine the magnetic field created by one conductor

To calculate the magnetic field (B) created by one conductor at a distance (r) from it, we can use Ampere's law. Let's consider a current I₁ flowing through the first conductor. The formula for the magnetic field due to this current is given by: B = \(\frac{\mu_0 I_1}{2 \pi r}\) where \(\mu_0\) is the permeability of free space. Now let's find the force acting on the second conductor due to this magnetic field.

Calculate the force on the second conductor

Now, we consider a current I₂ flowing through the second conductor. The force (F) on this conductor can be calculated using the formula: F = I₂ × (L × B) where L is the unit length of the conductor, which can be taken as 1 meter. Plugging in the formula for B from Step 1 into this equation, we get: F = I₂ × \(\frac{\mu_0 I_1}{2 \pi r}\) Now we can find the force per unit length.

Calculate the force per unit length

The force per unit length (f) can be found by dividing the force F by the unit length L (which is 1 meter in our case): f = \(\frac{F}{L}\) Since L = 1, we can state that: f = I₂ × \(\frac{\mu_0 I_1}{2 \pi r}\) Now let's analyze the graph of this force per unit length formula with respect to the distance between the conductors (r).

Analyze the graph of force per unit length between the conductors

The formula for force per unit length is given by: f = I₂ × \(\frac{\mu_0 I_1}{2 \pi r}\) We can observe that the force per unit length (f) is inversely proportional to the distance (r) between the conductors: f ∝ \(\frac{1}{r}\) This inverse relationship indicates that the graph shape should be a rectangular hyperbola. Therefore, the correct answer is (d) Rectangular hyperbola.