Problem 74 - Master Resource Book in JEE Main Physics

Question

Two long parallel wires \(P\) and \(Q\) are both perpendicular to the plane of the paper with distance \(5 \mathrm{~m}\) between them. If \(P\) and \(Q\) carry current of \(2.5 \mathrm{~A}\) and 5 A respectively in the same direction, then the magnetic field at a point half way between the wires is (a) \(\frac{\sqrt{3} \mu_{0}}{2 \pi}\) (b) \(\frac{\mu_{0}}{\pi}\) (c) \(\frac{3 \mu_{0}}{2 \pi}\) (d) \(\frac{\mu_{0}}{2 \pi}\)

Step-by-Step Solution

Answer

The magnetic field at the midpoint is \(\frac{\mu_{0}}{2\pi}\), so option (d) is correct.

Step 1: Understand the scenario

You have two parallel wires, both carrying current in the same direction with a distance of \(5\,\mathrm{m}\) between them. Wire \(P\) carries 2.5 A, and wire \(Q\) carries 5 A. You need to calculate the net magnetic field at the midpoint between the two wires.

Step 2: Calculate the magnetic field of wire P at the midpoint

Use the formula for the magnetic field generated by a long straight current-carrying wire: \(B = \frac{\mu_{0}I}{2 \pi r}\), where \(I\) is the current and \(r\) is the distance from the wire. At the midpoint, \(r = 2.5\,\mathrm{m}\) for both wires P and Q.For wire P: \[ B_{P} = \frac{\mu_{0} \times 2.5}{2 \pi \times 2.5} = \frac{\mu_{0}}{2 \pi} \]

Step 3: Calculate the magnetic field of wire Q at the midpoint

Similarly, calculate the magnetic field due to wire Q at the midpoint using the same formula.For wire Q: \[ B_{Q} = \frac{\mu_{0} \times 5}{2 \pi \times 2.5} = \frac{\mu_{0} \times 2}{2 \pi} = \frac{2\mu_{0}}{2 \pi} = \frac{\mu_{0}}{\pi} \]

Step 4: Determine the direction of each magnetic field

Since both currents flow in the same direction, at the midpoint, the direction of magnetic fields from both wires will be opposite. According to the right-hand rule, the magnetic field due to wire P will point into the page and due to wire Q will also point into the page. However, they oppose each other because they are spaced symmetrically.

Step 5: Calculate the net magnetic field at the midpoint

The net magnetic field is the difference because they are in opposite directions. Subtract the magnetic field due to wire P from wire Q:\[ B_{\text{net}} = B_{Q} - B_{P} = \frac{\mu_{0}}{\pi} - \frac{\mu_{0}}{2\pi} = \frac{\mu_{0}}{2\pi} \]

Step 6: Select the correct answer

Thus, the correct multiple-choice option that represents the magnetic field at the point halfway between the wires is (d) \(\frac{\mu_{0}}{2 \pi}\).

Key Concepts

Ampere's Law

Ampere's Law is a fundamental principle used to understand the creation of magnetic fields by electric currents. It is formally expressed as \( abla \times \mathbf{B} = rac{abla imes abla}{abla} = rac{4\pi \cdot \mu_0 \cdot I}{c^2} \), where \( \mu_0 \) is the magnetic permeability, \( \mathbf{B} \) is the magnetic field, and \( I \) is the current enclosed by a loop. This law is particularly useful for calculating magnetic fields in scenarios with a high degree of symmetry, like long straight wires or solenoids. For parallel wires, the law implies that the magnetic field circles around the current, forming concentric loops.When analyzing parallel current-carrying wires, the interaction between their fields can lead to attraction or repulsion, depending on the direction of their currents. In our exercise scenario, both wires run current in the same direction, indicating that the magnetic fields interact symmetrically around each wire. Ampere's Law informs us that this symmetrical setup helps determine that the magnetic contributions will oppose at any midpoint between the wires. Using this law simplifies the process of calculating the resultant magnetic force produced at any given point, such as halfway between the wires.

Biot-Savart Law

The Biot-Savart Law is central to understanding the magnetic field generated by a small segment of current-carrying wire. The formula for this law is given by \( dB = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \hat{\mathbf{r}}}{r^2} \), where \( dB \) is the infinitesimal contribution to the magnetic field, \( I \) is the current through the wire, \( d\mathbf{l} \) is a segment of the wire, \( \hat{\mathbf{r}} \) is the unit vector pointing from the wire segment to the point of interest, and \( r \) is the distance from the wire to the point.In simpler terms, the Biot-Savart Law helps us understand how small segments of a wire contribute individually to the total magnetic field at a point. This law is especially useful in scenarios where Ampere's Law might be less applicable due to geometrical complexities. In our problem, though it may not be directly applied, it gives a foundational understanding of how magnetic fields form around current-carrying wires and how these fields can be summed to get a total field, contributing to the magnetic fields calculated from parallel wires.

Right-hand rule

The right-hand rule is a mnemonic for determining the direction of the magnetic field relative to a current's direction. To apply the rule, point your right thumb in the direction of the current and curl your fingers around the wire. Your fingers will naturally wrap in the direction that the magnetic field lines encircle the wire. This rule is particularly helpful in our case of parallel wires. For each wire, the right-hand rule aids in visualizing the orientation of the magnetic fields around them. When we consider a point midway between these two wires, the right-hand rule shows us that although the currents in both wires flow in the same direction, their respective magnetic fields will oppose each other at the midpoint. This antiparallel orientation is crucial when calculating the net magnetic field, as observed in the exercise, resulting in the cancellation of some magnetic influences and the subtraction of their magnitudes.

Magnetic field due to current

A current flowing through a wire generates a magnetic field, an established principle derived from the interplay of electricity and magnetism. The strength of this magnetic field at a certain distance from the wire can be calculated using the formula \( B = \frac{\mu_0 I}{2 \pi r} \). Here, \( B \) is the magnetic field strength, \( \mu_0 \) is the magnetic constant, \( I \) is current strength, and \( r \) is the distance from the wire.For the problem at hand, we apply this formula to each parallel wire separately. At the midpoint between the two wires, which is \( 2.5 \, \mathrm{m} \) away from each, we calculate their individual contributions to the magnetic field. As per the solution, the fields oppose each other because they have currents in the same direction, affecting their magnetic interactions at the midpoint. The net magnetic field is a simple arithmetic difference of the magnitudes, accounted for by assessing that the fields encircle the wires in opposing loops, when viewed from their respective reference points.