Question

The forces existing between two parallel current carrying conductors is \(\mathrm{F}\). If the current in each conductor is doubled, then the value of force will be (a) \(2 \mathrm{~F}\) (b) \(4 \mathrm{~F}\) (c) \(5 \mathrm{~F}\) (d) \((\mathrm{F} / 2)\)

Short answer

The short answer to the question is: If the current in each conductor is doubled, the new value of the force between the two parallel current carrying conductors will be: \( 4 \mathrm{~F} \)

Step-by-step solution

Write the formula for the force between parallel conductors

The formula for the force per unit length (f) between two long parallel conductors with currents I₁ and I₂ separated by a distance d is given by: \( f = \dfrac{\mu_0 I_1 I_2}{2 \pi d} \) Where: \( \mu_0 \) is the permeability of free space (approximately \( 4 \pi × 10^{−7} \) T⋅m/A), I₁ and I₂ are the currents in the conductors, and d is the distance between the conductors.

State the given values and the values of the doubled current

The given force between the conductors is F. Let's denote the original current in each of the conductors as \( I \). When the currents are doubled, the new current values will be \( 2I \) in each conductor.

Write the formula for the force after doubling the current

Now, to find the force between the conductors after the currents are doubled, we can use the same formula, substituting the doubled current values \( 2I \) for I₁ and I₂: \( f' = \dfrac{\mu_0 (2I) (2I)}{2 \pi d} \)

Relate the original force with the new force to find the change in force.

To find the change in force, we need to relate the original force (F) to the new force (f'). We can do this by dividing the second equation by the first equation: \( \frac{f'}{f} = \frac{\dfrac{\mu_0 (2I) (2I)}{2 \pi d}}{\dfrac{\mu_0 I I}{2 \pi d}} \) Now, let's simplify the equation to find the change in force: \( \Rightarrow \frac{f'}{f} = \frac{(2I)(2I)}{I I} \) \( \Rightarrow \frac{f'}{f} = 4 \) This tells us that the new force \( f' \) is 4 times the original force (F).

Select the correct answer from the options given.

The correct answer for the value of the force when the currents in both conductors are doubled is: (b) \( 4 F \)