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Chapter 15: Problem 157

Two long conductors, separated by a distance \(d\), carry current \(I_{1}\) and \(I_{2}\) in the same direction. They exert a force \(F\) on each other. Now the current in one of them is doubles and its direction is reversed. The distance is also increased to \(3 d\). The new value of the force between them is (A) \(-\frac{2 F}{3}\) (B) \(\frac{F}{3}\) (C) \(-2 F\) (D) \(-\frac{F}{3}\)

Short Answer

Expert verified
The new value of the force between the conductors is (A) \(-\frac{2 F}{3}\).

Step by step solution

01

Determine Ampere's Law

Ampere's Law (also known as the Biot-Savart Law) describes the force between two conductive wires carrying currents. The formula is: \[F = \frac{\mu_{0} I_{1} I_{2} L}{2\pi d}\] Here, \(F\) is the force between the conductors, \(\mu_0\) is the permeability of free space, \(I_1\) and \(I_2\) are the currents in the conductors, \(L\) is the length common to both conductors, and \(d\) is the distance between the conductors. Step 2: Modify the parameters based on given conditions
02

Update the given conditions

We are given that the current in one of the conductors doubles, its direction is reversed, and the distance between them increases to \(3d\). Let's assume that the current \(I_1\) changes. The new parameters are: - The current in conductor 1 becomes \(-2I_1\) - The current in conductor 2 remains the same, which is \(I_2\) - The distance between the conductors now becomes \(3d\) Step 3: Calculate the new force between the conductors
03

Apply updated parameters to Ampere's Law

Now we need to apply the modified parameters to Ampere's Law and find the updated force: \[F' = \frac{\mu_{0} (-2I_{1}) I_{2} L}{2\pi (3d)}\] Step 4: Find the relation between the initial and updated forces
04

Calculate the ratio of the updated force to the initial force

In order to find the relation between the initial and updated forces, we can divide the equation for \(F'\) by the equation for \(F\): \[\frac{F'}{F} = \frac{\frac{\mu_{0} (-2I_{1}) I_{2} L}{2\pi (3d)}}{\frac{\mu_{0} I_{1} I_{2} L}{2\pi d}} = -\frac{2}{3}\] Step 5: Identify the answer
05

Compare the relation to the given options

The relation between the updated force and initial force is \(-\frac{2}{3}\). Which means: \[F' = -\frac{2F}{3}\] So, the correct answer is (A) \(-\frac{2 F}{3}\).

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A uniform electric field and a uniform magnetic field are acting along the same direction in a certain region. If an electron is projected along the direction of the fields with a certain velocity then \([\mathbf{2 0 0 5}]\) (A) its velocity will increase. (B) its velocity will decrease. (C) it will turn towards left of direction of motion. (D) it will turn towards right of direction of motion.

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