Problem 11 - Master Resource Book in JEE Main Physics

Question

An element, \(d l=d x\) \hat{ i } (where \(d x=1 \mathrm{~cm}\) ) is placed at the origin and carries a large current \(i=10 \mathrm{~A}\). What is the magnetic field on the \(y\)-axis at a distance of \(0.5 \mathrm{~m} ?\) (a) \(2 \times 10^{-8} \hat{\mathrm{k}} \mathrm{T}\) (b) \(4 \times 10^{-8} \hat{\mathrm{k}} \mathrm{T}\) (c) \(-2 \times 10^{-8} \hat{\mathrm{k}} \mathrm{T}\) (d) \(-4 \times 10^{-8} \hat{\mathrm{k}} \mathrm{T}\)

Step-by-Step Solution

Answer

The magnetic field is \(4 \times 10^{-8} \hat{\mathbf{k}} \text{ T}\) (option b).

Step 1: Understanding the Problem

We are asked to find the magnetic field on the y-axis at a certain distance from a current-carrying element. The element carries a large current of 10 A and extends in the x-direction by 1 cm. We need to find the field at 0.5m on the y-axis using the Biot-Savart Law.

Step 2: Apply the Biot-Savart Law

The Biot-Savart Law for a current element can be given as \( d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{i \, d\mathbf{l} \times \mathbf{r}}{r^3} \) where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \), \( i = 10 \, \text{A} \), \( d\mathbf{l} = dx \hat{\mathbf{i}} \), and \( \mathbf{r} \) is the position vector from the current element to the point where the field is being calculated.

Step 3: Determine Vector Quantities

The vector \( d\mathbf{l} = dx \hat{\mathbf{i}} \) and the position vector \( \mathbf{r} \) from the origin to a point \(0.5 \, \text{m}\) on the y-axis is \( \mathbf{r} = 0.5 \hat{\mathbf{j}} \) m. Thus, \( r = 0.5 \, \text{m} \). The cross product is \( d\mathbf{l} \times \mathbf{r} = dx \hat{\mathbf{i}} \times 0.5 \hat{\mathbf{j}} = dx \times 0.5 \hat{\mathbf{k}} \).

Step 4: Calculate the Magnetic Field

Substitute the known values into the Biot-Savart Law:\[ dB = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{10 \times 0.01 \times 0.5}{(0.5)^3} \hat{\mathbf{k}} \]Simplify the expression:\[ dB = 10^{-7} \cdot \frac{0.005}{0.125} \hat{\mathbf{k}} \]\[ dB = 4 \times 10^{-8} \hat{\mathbf{k}} \text{ T} \]

Step 5: Identify the Direction

The right-hand rule for the cross product \( dx \hat{\mathbf{i}} \times 0.5 \hat{\mathbf{j}} \) gives a direction in the positive \( \hat{\mathbf{k}} \) direction. Thus, the magnetic field is in the positive \( \hat{\mathbf{k}} \) direction.

Key Concepts

Magnetic Field

The magnetic field is a vector field that surrounds electric currents and magnetic dipoles. It exerts a force on moving charges and other magnetic entities. In this context, the magnetic field is generated by a current element, which is a small segment of a current-carrying conductor.

Magnetic fields are typically measured in teslas (T) and can be visualized as lines of force extending indefinitely in three-dimensional space. The strength and direction of the magnetic field depend on several factors, including the magnitude of the current, the direction of the current, and the positions in relation to the current element.

• Strength: Proportional to the current and inversely proportional to the square of the distance from the current.
• Direction: Given by the point of interest with respect to the current's position.

Understanding this helps determine how the magnetic field is oriented around the element and its behavior at various points in space.

Current Element

A 'current element' refers to a small segment of a conductor through which current flows. It's typically represented as a vector (\( doldsymbol{l} \)) that indicates both the magnitude of the current and its direction. In problems related to magnetic fields, such as the above exercise, current elements are crucial, as they help calculate the resultant magnetic field due to their interaction with the surrounding space.

In our task, the current is flowing along the x-direction for a minuscule segment (\( doldsymbol{l} = dx oldsymbol{ ext{î}} \)), which is crucial for solving the magnetic field direction and value. It highlights how these segments interact with points in space to influence the magnetic field vectors formed along axes like the y-axis.

Cross Product

The cross product is a mathematical operation on two vectors that yields a third vector perpendicular to the plane of the initial vectors. It's a fundamental tool in calculating fields in physics, including magnetic fields. For a current element, using Biot-Savart Law, the cross product is vital for determining the direction of the magnetic field.

In our exercise, the Biot-Savart Law employs the cross product of the current element vector (\( doldsymbol{l} \)) and the radial vector (\( oldsymbol{r} \)) to derive the magnetic field (\( oldsymbol{dB} \)).

• The vectors are perpendicular, and the magnitude of the cross product is maximized when these vectors are perpendicular.
• The resulting vector direction is given by the right-hand rule.

The cross product thus directly influences both the strength and direction of the magnetic field generated around the current element.

Right-Hand Rule

The right-hand rule is a simple method used in physics to determine the direction of certain vector operations, such as the cross product. In the context of the magnetic field, it provides an intuitive way to discern the orientation of the field lines based on the direction of currents and their configurations.

To apply the right-hand rule for a cross product in the Biot-Savart Law:

• Point your thumb in the direction of the current element (\( doldsymbol{l} \)).
• Extend your fingers in the direction of the position vector (\( oldsymbol{r} \)).
• Your palm will point in the direction of the magnetic field (\( oldsymbol{dB} \)), perpendicular to both vectors.

This rule is an indispensable tool for visualizing the interactions between vectors in spatial dimensions and ensuring accurate calculations of field directions.