Problem 6 - Master Resource Book in JEE Main Physics

Question

A square of side \(L\) metres lies in the x-y plane in a region. Where the magnetic field is given by \(\mathbf{B}=B_{0}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathbf{T}\), where \(B_{0}\) is constant. The magnitude of flux passing through the square is [NCERT Exemplar] (a) \(2 B_{0} L^{2} \mathrm{~Wb}\) (b) \(3 B_{0} L^{2} \mathrm{~Wb}\) (c) \(4 B_{0} L^{2}\) Wb (d) \(\sqrt{29} B_{0} L^{2} \mathrm{~Wb}\)

Step-by-Step Solution

Answer

The magnetic flux is 4B_0 L^2 Wb. Choice (c) is correct.

Step 1: Understand the Problem

We are given a magnetic field B = B_{0}(2 i + 3 j + 4 k) T and a square of side length L lying in the xy-plane. Our task is to compute the magnetic flux through the square.

Step 2: Recall the Flux Formula

The formula for the magnetic flux f is given by the surface integral of the magnetic field over the area, which simplifies to f = B cdot A, where A is the area vector perpendicular to the surface.

Step 3: Determine the Area Vector

Since the square lies in the xy-plane, the area vector A is perpendicular to the plane and has a magnitude equal to the area of the square: L^2. Its direction is along k-axis (z-axis), so A = L^2 k.

Step 4: Calculate the Dot Product

Compute the dot product of B and A:  (B_{0}(2 i + 3 j + 4 k)) cdot (L^2 k) = B_{0} L^2 (0 + 0 + 4) = 4B_{0} L^2.

Step 5: Choose the Correct Answer

The calculated magnetic flux is 4B_{0} L^2 Wb. Corresponding option is (c) 4B_{0} L^{2}.

Key Concepts

Magnetic Field

The concept of a **magnetic field** is an essential part of understanding physical phenomena related to magnets and electromagnetic interactions. A magnetic field can be thought of as a region around a magnet within which magnetic forces are exerted.
This field is often represented by the symbol \( \mathbf{B} \), and it has both magnitude and direction. In the given problem, the magnetic field \( \mathbf{B} \) is specified as \( B_{0}(2 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}} + 4 \hat{\mathbf{k}}) \). This expression tells us how the magnetic field is oriented in space.

• \( 2 \hat{\mathbf{i}} \) indicates the contribution in the x-direction.
• \( 3 \hat{\mathbf{j}} \) indicates the contribution in the y-direction.
• \( 4 \hat{\mathbf{k}} \) indicates the contribution in the z-direction.

The constant \( B_0 \) scales the overall strength of the magnetic field.

Surface Integral

The **surface integral** is a mathematical tool needed to calculate things like magnetic flux through a surface. In simple terms, it adds up the values over a surface area. This is crucial when determining how much of a vector field, like a magnetic field, penetrates through a specified surface.When we use a surface integral to calculate magnetic flux, we are looking at how the magnetic field interacts with the surface. For a flat surface like a square, this simplifies into the much simpler dot product of the magnetic field and the area vector.In mathematical terms, the surface integral for magnetic flux \( \Phi \) through a surface is given by:\[ \Phi = \int \mathbf{B} \cdot d\mathbf{A} \] When the surface is uniform and the magnetic field is constant over this surface, this integral simplifies to a dot product, streamlining the calculation process.

Area Vector

An **area vector** is a simple yet fascinating concept. For any flat surface, the area vector is perpendicular to the surface itself and has a magnitude equal to the area of the surface. Here, since the square lies within the xy-plane, the area vector \( \mathbf{A} \) is directed along the z-axis (perpendicular to the xy-plane), represented as:\[ \mathbf{A} = L^2 \hat{\mathbf{k}} \] This means the vector points directly 'out of' the plane of the square. The direction along \( \hat{\mathbf{k}} \) signifies the perpendicular nature relating to the xy-plane. It's like imagining that vector protruding from the center of the square towards you!

Dot Product

Understanding the **dot product** simplifies finding how much of a vector passes through another vector or surface. It is a mathematical operation that combines two vectors, giving us a scalar, which tells us how much one vector goes in the direction of another.In the context of flux calculations, the dot product helps to find the relationship between the magnetic field and the area vector. The formula for the dot product of vectors \( \mathbf{B} = (B_x \hat{i} + B_y \hat{j} + B_z \hat{k}) \) and \( \mathbf{A} = (A_x \hat{i} + A_y \hat{j} + A_z \hat{k}) \) is:\[ \mathbf{B} \cdot \mathbf{A} = B_x A_x + B_y A_y + B_z A_z \]For the given problem, since the area vector \( \mathbf{A} = L^2 \hat{\mathbf{k}} \) only has a component in the z-direction, we only consider the component of \( \mathbf{B} \) in the same direction:\[ \mathbf{B} \cdot \mathbf{A} = (0) + (0) + (4B_0)(L^2) = 4B_0 L^2 \] This makes calculations straightforward, telling us that only the z-component contributes to the flux through the square surface.