Problem 17 - Master Resource Book in JEE Main Physics

Question

An infinitely long wire carrying current \(i\) is along \(Y\)-axis such that its one end is at point \((0, b)\) while the wire extends upto \(\infty .\) The magnitude of magnetic field strength at point \(P(a, 0)\) is (a) \(\frac{\mu_{0} i}{4 \pi a}\left(1+\frac{b}{\sqrt{a^{2}+b^{2}}}\right)\) (b) \(\frac{\mu_{0} i}{4 \pi a}\left(1-\frac{b}{\sqrt{a^{2}+b^{2}}}\right)\) (c) \(\frac{\mu_{0} i}{4 \pi a}\left(1-\frac{a}{\sqrt{a^{2}+b^{2}}}\right)\) (d) \(\frac{\mu_{0} i}{4 \pi a}\left(1+\frac{a}{\sqrt{a^{2}+b^{2}}}\right)\)

Step-by-Step Solution

Answer

The correct option is (b): \(\frac{\mu_{0} i}{4 \pi a}\left(1 - \frac{b}{\sqrt{a^2 + b^2}}\right)\).

Step 1: Understand the problem setup

We have an infinitely long wire carrying a current \(i\) along the \(Y\)-axis with one end at \((0, b)\) and extending to infinity upwards. We need to determine the magnetic field strength at point \(P(a, 0)\).

Step 2: Use Biot-Savart Law

The magnetic field \(dB\) produced by a small segment \(dl\) of the wire at a distance \(r\) is given by the Biot-Savart law: \(dB = \frac{\mu_{0}}{4\pi} \cdot \frac{i \, dl \times \mathbf{r}}{r^3}\), where \(\mathbf{r}\) is the position vector from the wire segment to point \(P\).

Step 3: Set up coordinate system and positions

Consider a differential element along the wire at position \((0, y)\). The position vector from this element to point \(P(a, 0)\) is \(\mathbf{r} = (a, 0) - (0, y) = (a, -y)\).

Step 4: Calculate distance from wire to point P

The distance from the differential segment \((0, y)\) to point \((a, 0)\) is \(r = \sqrt{a^2 + y^2}\).

Step 5: Determine the cross product direction

Since the wire is along the \(Y\)-axis, the cross product \(dl \times \mathbf{r}\) points in the \(-Z\)-direction due to the right-hand rule, which matches the typical direction of magnetic fields created by a current. The direction \(\hat{k}\).

Step 6: Integrate over the length of the wire

To find the total magnetic field at \(P\), integrate \(dB\) from \(y = b\) to \(y = \infty\). This gives \(B = \frac{\mu_0 i}{4 \pi} \int_b^{\infty} \frac{dl \cdot a}{(a^2 + y^2)^{3/2}}\), where \(dl = dy\).

Step 7: Evaluate the integral

Perform the integration: \(\int_b^{\infty} \frac{dy}{(a^2 + y^2)^{3/2}} = \left[\frac{y}{a^2 \sqrt{a^2 + y^2}}\right]_b^{\infty}\). As \(y\) goes to infinity, the term approaches 0. Evaluate at \(b\): the magnetic field at \(P\) becomes \(B = \frac{\mu_{0} i}{4 \pi a}\left(\frac{b}{\sqrt{a^2 + b^2}} - 1\right)\).

Step 8: Choose the correct answer

Based on our calculations, the correct option is \(\frac{\mu_{0} i}{4 \pi a}\left(1 - \frac{b}{\sqrt{a^2 + b^2}}\right)\), which corresponds to option (b).

Key Concepts

Magnetic field due to a current

In physics, the magnetic field created by an electric current is a fundamental concept that is explained using the Biot-Savart Law. This principle helps us understand the behavior of magnetic fields around wires carrying current. When current flows through a conductor like a wire, it generates a magnetic field in the surrounding space. The direction and strength of this field depend on several factors, including the magnitude of the current and the geometry of the wire.

The Biot-Savart Law provides a quantitative description of these relationships. It states that a small segment of current-carrying wire, called a differential element, produces a magnetic field at a specific point in space. This field is proportional to the amount of current in the segment and inversely proportional to the square of the distance from the segment to the point of interest.

The law is mathematically expressed as:

• \( dB = \frac{\mu_{0}}{4\pi} \cdot \frac{i \cdot dl \times \mathbf{r}}{r^3} \)
  

where \( dB \) is the magnetic field, \( \mu_{0} \) is the permeability of free space, \( i \) is the current, \( dl \) is the differential element of the wire, and \( \mathbf{r} \) is the position vector from the wire segment to the point.

Understanding the magnetic field due to a current is crucial for solving many physics problems, as it applies to numerous practical situations, from the tiny currents in microchips to large power lines.

Integration in physics

Integration is a powerful mathematical tool used in physics to sum up small contributions over a continuous range. In the context of finding the magnetic field using the Biot-Savart Law, integration helps us to calculate the total magnetic field produced by a current-carrying wire.

Considering an infinitely long wire, the entire length of the wire should be taken into account when determining the magnetic field at a specific point. This involves breaking the wire into many small elements, each contributing a small part of the magnetic field. The Biot-Savart Law calculates this part for each element, and then these contributions are summed up using integration over the entire extent of the wire.

In our example, integration is used to determine the entire field generated by the wire from the point where it starts (at \( y = b \)) and extends infinitely upwards. The integral of interest is:

• \( B = \frac{\mu_0 i}{4 \pi} \int_b^{\infty} \frac{a \, dy}{(a^2 + y^2)^{3/2}} \)

Integration assesses the cumulative effect of each infinitesimal contribution over the infinite length of the wire, resulting in the total magnetic field at the point of interest.

This technique is foundational in physics, facilitating precise and accurate solutions to complex problems. Whether calculating fields, forces, or energy, integration is indispensable for transitioning from understanding simple systems to more intricate scenarios.

Right-hand rule

The right-hand rule is a convention used to determine the direction of vectors in relation to rotation or fields. It’s particularly valuable when dealing with magnetic fields produced by currents. This rule assists in visualizing the orientation of magnetic fields, angular momentum, and other vector quantities as a result of currents and rotations.

To apply the right-hand rule in our case, imagine gripping the wire with your right hand, with your thumb pointing in the direction of the conventional current (pointing upwards along the \(Y\)-axis in this scenario). Your fingers will naturally curl around the wire, indicating the direction of the magnetic field encircling the wire.

Now, when it's about determining the direction of the vector product such as the cross product in Biot-Savart Law – \( dl \times \mathbf{r} \) – the right-hand rule is used once again. Here’s how you do it:

• Align your fingers in the direction of the first vector, \( dl \) (the current element along the wire, pointing upwards).
• Turn your palm to sweep towards the second vector, \( \mathbf{r} \) (from the wire to the point \(P(a, 0)\)).
• Your thumb now points in the direction of the resulting magnetic field vector, which will be in the \(-Z\)-direction.

Employing the right-hand rule simplifies understanding of the spatial relationships in vector mathematics, which becomes indispensable when analyzing electromagnetic phenomena such as magnetic fields.