Problem 97 - Master Resource Book in JEE Main Physics

Question

In a region of space, the electric field is given by \(\mathbf{E}=8 \hat{\mathbf{i}}+` 4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\). The electric flux through a surface of area of 100 units \(x y\)-plane is (a) 800 units (b) 300 units (c) 400 units (d) 1500 units

Step-by-Step Solution

Answer

The electric flux is 300 units, option (b).

Step 1: Understand the Problem

We are given an electric field vector \( \mathbf{E}=8 \hat{\mathbf{i}}+ 4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} \) and a surface within the \(xy\)-plane with an area of 100 units. We are asked to find the electric flux through this surface and then determine which of the given options corresponds to it.

Step 2: Define Electric Flux Equation

The electric flux \( \Phi \) through a surface is given by the dot product of the electric field \( \mathbf{E} \) and the area vector \( \mathbf{A} \). The equation is \( \Phi = \mathbf{E} \cdot \mathbf{A} \). For a flat surface in the \(xy\)-plane, the area vector is perpendicular to the \(xy\)-plane, in the \(\hat{\mathbf{k}}\) direction, and has a magnitude equal to the area of the surface (100 units). Hence, \( \mathbf{A} = 100 \hat{\mathbf{k}} \).

Step 3: Compute the Dot Product

Compute the dot product \( \mathbf{E} \cdot \mathbf{A} \). Recall that the dot product between vectors \( \mathbf{a} = a_x \hat{\mathbf{i}} + a_y \hat{\mathbf{j}} + a_z \hat{\mathbf{k}} \) and \( \mathbf{b} = b_x \hat{\mathbf{i}} + b_y \hat{\mathbf{j}} + b_z \hat{\mathbf{k}} \) is given by \( a_x b_x + a_y b_y + a_z b_z \). Since \( \mathbf{A} = 100 \hat{\mathbf{k}} \), the dot product becomes: \[ \Phi = 8 \times 0 + 4 \times 0 + 3 \times 100 = 300 \] since the \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) components are zero for \( \mathbf{A} \).

Step 4: Find Corresponding Option

The computed electric flux \( \Phi \) is 300 units. Now compare this with the provided options: (a) 800 units, (b) 300 units, (c) 400 units, (d) 1500 units. The electric flux matches option (b), which is 300 units.

Key Concepts

Electric Field

An electric field is a physical field that surrounds electrically charged particles and exerts forces on other charged particles within the field. The electric field is a vector quantity, meaning it has both magnitude and direction. In this exercise, we are dealing with an electric field expressed as the vector \( \mathbf{E}=8 \hat{\mathbf{i}}+ 4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} \). Here, the components along each axis (\(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\), \(\hat{\mathbf{k}}\)) represent the strength of the field in the respective coordinate directions:

• \( 8 \hat{\mathbf{i}} \) indicates that there is a significant component of the electric field directed along the x-axis.
• \( 4 \hat{\mathbf{j}} \) represents a moderate field component along the y-axis.
• \( 3 \hat{\mathbf{k}} \) shows a lesser component directed along the z-axis.

Understanding the components of an electric field is important as it influences how the field can interact with charges, and these interactions vary in different directions. Any surface placed within the electric field will experience a flux depending on its orientation to these components.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a single scalar quantity. It is invaluable in vector calculus for calculations involving angles, projections, and determining perpendicularity. The formula for the dot product of vectors \( \mathbf{a} = a_x \hat{\mathbf{i}} + a_y \hat{\mathbf{j}} + a_z \hat{\mathbf{k}} \) and \( \mathbf{b} = b_x \hat{\mathbf{i}} + b_y \hat{\mathbf{j}} + b_z \hat{\mathbf{k}} \) is \[ \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z.\] In the context of electric flux through a surface, the dot product is used to calculate how much of the electric field passes through the surface. The area vector \( \mathbf{A} \) is perpendicular to the surface, meaning it indicates the direction in which the surface is oriented.
For a surface within the xy-plane, the area vector is typically represented entirely by \( \hat{\mathbf{k}} \) because it is perpendicular to the plane. In our exercise, this simplifies the dot product calculation significantly because:

• The x and y components of the area vector are zero.
• Only the z component (\( a_z = 100 \)) is used, thus greatly simplifying the computation of electric flux.

This simplification exemplifies the practical utility of the dot product in physics and engineering calculations. Calculating the dot product gives us the electric flux, which determines how strongly the electric field interacts with the surface.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vectors and operations on vectors in multiple dimensions. It combines algebraic and calculus concepts to analyze vector fields, which are essential in various physical systems. Electric fields, magnetic fields, fluid flow, and many other physical phenomena can be described as vector fields, where every point in space is associated with a vector. These fields are analyzed using vector calculus, allowing us to understand and predict interactions within the field. Some fundamental operations in vector calculus include:

• Gradient: Measures the rate and direction of change in a scalar field, producing a vector field.
• Divergence: Provides a scalar output indicating how much a vector field spreads outward from a point.
• Curl: Produces a vector that describes the rotation of a vector field around a point.
• Integral operations: Such as line, surface, and volume integrals, are used to analyze sums over paths, areas, or volumes.

In the context of electric flux, which involves the dot product of an electric field vector with an area vector, vector calculus helps in understanding how field lines interact with surfaces. Calculating flux is more straightforward when using vector calculus, making complex physical phenomena easier to analyze. The formulations within vector calculus provide the essential tools to manipulate and understand the behavior of vector fields under various conditions.