Problem 4 - Master Resource Book in JEE Main Physics

Question

Two very long, straight, parallel wires carry steady currents \(i\) and \(-i\) respectively. The distance between the wires is \(d\). At a certain instant of time, a point charge \(q\) is at a point equidistant from the two wires, in the plane of the wires. Its instantaneous magnitude of the force due to the magnetic field acting on the charge at this instant is (a) \(\frac{\mu_{0} i q v}{2 \pi d}\) (b) \(\frac{\mu_{0} i q v}{\pi d}\) (c) \(\frac{2 \mu_{0} i q v}{\pi d}\) (d) zero

Step-by-Step Solution

Answer

The instantaneous magnitude of the force is zero (option d).

Step 1: Understand the Problem

Two parallel wires are carrying currents of equal magnitude but in opposite directions. A point charge is placed equidistant from both wires in the plane of the wires. We need to calculate the magnetic force acting on this charge.

Step 2: Magnetic Field Created by a Single Wire

The magnetic field created by a wire carrying current can be calculated using Ampère's Law: \[ B = \frac{\mu_0 i}{2 \pi r} \]where \( r \) is the distance from the wire.

Step 3: Calculate the Distance of Charge from Wires

Since the charge is equidistant from both wires and in the plane of the wires, its distance from each wire is \( d/2 \).

Step 4: Determine the Direction of the Magnetic Fields

Using the right-hand rule: For the top wire with current \(i\), the magnetic field at the charge is directed into the plane. For the bottom wire with current \(-i\), the magnetic field is directed out of the plane.

Step 5: Calculate Resultant Magnetic Field

Since the magnetic fields due to each wire have the same magnitude but opposite directions at the charge's position, they cancel each other out, resulting in a net magnetic field of zero at the position of the charge.

Step 6: Calculate Magnetic Force on Charge

The magnetic force acting on a charge \( q \) moving with velocity \( v \) is given by:\[ F = q(v \times B) \]Since the net magnetic field \( B = 0 \), the force \( F = 0 \).

Key Concepts

Ampère's Law

Ampère's Law is a fundamental tool in electromagnetism used to relate magnetic fields to the currents that produce them. It states that for any closed loop path, the sum of the magnetic field (B) along the path is equal to the permeability of free space (μ₀) times the current (i) enclosed by the path. This is mathematically expressed as: \[ \oint B \cdot dl = \mu_0 I \] In simpler terms, it means that current flowing through a wire generates a magnetic field around it. The strength and direction of this magnetic field depend on the amount of current and the distance from the wire. Ampère's Law is especially useful for calculating the magnetic fields in symmetric situations like inside a solenoid, around a loop of current, or between parallel wires.

Right-Hand Rule

The right-hand rule is a handy shortcut for determining the direction of the magnetic field relative to the current flow. To use it, point your thumb in the direction of the current, and your fingers will curl in the direction of the magnetic field lines. It is especially useful in problems involving straight wires and loops.

• Your thumb points in the direction of conventional current (positive to negative).
• Your fingers represent the circular magnetic field lines encircling the wire.

This rule simplifies the process of finding out which way the magnetic field is oriented. In the context of two parallel wires, it helps in visualizing that the magnetic fields produced by opposing currents will be in opposite directions, leading to the possibility of them canceling each other out at certain points.

Parallel Wires Current

When dealing with parallel wires that carry currents, an interesting phenomenon occurs. They either attract or repel each other, depending on the direction of the currents. Two wires with parallel currents flowing in the same direction attract each other. Conversely, if the currents are flowing in opposite directions, the wires repel one another.
These forces arise because of the interaction between the magnetic fields generated by each wire. For long, straight wires, the magnetic field due to one wire exerts a force on the other. When the wires are placed at a distance d apart, the force per unit length can be calculated using:\[ F/L = \frac{μ_0 i_1 i_2}{2πd} \]In the exercise, since the currents are equal in magnitude but opposite in direction, the magnetic fields they produce around a midpoint effectively cancel each other. Thus, the force exerted on a charge placed equidistant from the wires is zero.

Magnetic Field Calculation

Calculating the magnetic field generated by a current-carrying wire is a key step in solving electromagnetic problems. According to Ampère's Laws, the magnetic field (B) at a distance r from a long, straight wire carrying a steady current (i) is given by:\[ B = \frac{μ_0 i}{2πr} \]Here, \(μ_0\) is the permeability of free space, and \(r\) is the radial distance from the wire. The formula shows that the magnetic field strength decreases as we move further away from the wire.In the given exercise, the point charge is equidistant from both wires, effectively making \(r = d/2\). Due to the opposite directions of the current in the wires, the net magnetic field experienced by the charge at that point is zero. This leads to zero magnetic force acting on the charge, according to the Lorentz Force Law, \( F = q(v \times B) \). As the magnetic field \(B\) is zero, so is the force \(F\).