Problem 130 - Master Resource Book in JEE Main Physics

Question

An electric charge \(+q\) moves with velocity \(\mathbf{v}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+\hat{\mathbf{k}}\), in an electromagnetic field given by \(\mathbf{E}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \mathbf{B}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}} .\) The \(y\)-component of the force experienced by \(+q\) is (a) \(2 q\) (b) \(11 q\) (c) \(5 q\) (d) \(3 q\)

Step-by-Step Solution

Answer

The y-component of the force is \(11q\).

Step 1: Identify the formula for calculating force

To find the force experienced by a charge in both electric and magnetic fields, we use the Lorentz force formula: \[\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\]where \(\mathbf{F}\) is the force, \(q\) is the charge, \(\mathbf{E}\) is the electric field, \(\mathbf{v}\) is the velocity of the charge, and \(\mathbf{B}\) is the magnetic field.

Step 2: Calculate the cross product \(\mathbf{v} \times \mathbf{B}\)

To find \(\mathbf{v} \times \mathbf{B}\), we compute the determinant:\[\begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ 3 & 4 & 1 \ 1 & 1 & -3 \end{vmatrix}.\]Calculating this determinant gives us:\[\mathbf{v} \times \mathbf{B} = \left| \begin{array}{ccc}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \3 & 4 & 1 \1 & 1 & -3 \\end{array} \right| = \hat{\mathbf{i}}(4(-3) - 1 \cdot 1) - \hat{\mathbf{j}}(3 \cdot (-3) - 1 \cdot 1) + \hat{\mathbf{k}}(3 \cdot 1 - 4 \cdot 1).\]This simplifies to:\[\mathbf{v} \times \mathbf{B} = -13\hat{\mathbf{i}} + 10\hat{\mathbf{j}} + (-1)\hat{\mathbf{k}} = -13\hat{\mathbf{i}} + 10\hat{\mathbf{j}} - \hat{\mathbf{k}}.\]

Step 3: Calculate the total force \(\mathbf{F}\)

Using the Lorentz force formula, substitute \(\mathbf{E}\) and the cross product result:\[\mathbf{F} = q((3\hat{\mathbf{i}} + \hat{\mathbf{j}} + 2\hat{\mathbf{k}}) + (-13\hat{\mathbf{i}} + 10\hat{\mathbf{j}} - \hat{\mathbf{k}}))\]Simplifying gives:\[\mathbf{F} = q((-10\hat{\mathbf{i}}) + 11\hat{\mathbf{j}} + \hat{\mathbf{k}}).\]

Step 4: Extract the y-component of the force

From the total force vector \(\mathbf{F} = q((-10\hat{\mathbf{i}}) + 11\hat{\mathbf{j}} + \hat{\mathbf{k}})\), the \(y\)-component is \(11q\).

Key Concepts

Electromagnetic Field

An electromagnetic field is a combination of electric and magnetic fields interacting in space. The electric field, denoted as \(\mathbf{E}\), describes the effect of electric charges at rest. It determines the force experienced by a unit charge present in its vicinity. Alternatively, the magnetic field, denoted as \(\mathbf{B}\), arises from moving electric charges (or currents) and affects other moving charges in the field.
It's important to remember:

• The electric field \(\mathbf{E}\) is a vector, meaning it has both magnitude and direction.
• The magnetic field \(\mathbf{B}\) is also a vector.

Thus, the interaction between a charge moving with a velocity in both fields is quantified using the Lorentz force. This force is a crucial concept in electromagnetism, allowing us to understand how charged particles behave in these fields. The formula used is \(\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\), where \(\mathbf{F}\) is the force experienced by the charge, \(q\) is the charge, \(\mathbf{E}\) is the electric field, \(\mathbf{v}\) is the velocity, and \(\mathbf{B}\) is the magnetic field.
The Lorentz force equation tells us how these fields interact with moving charges, affecting their path and motion. While the electric field component directly interacts with the charge, providing a force in the same direction, the magnetic field contributes to a force that is perpendicular to both the magnetic field direction and the velocity of the charge.

Cross Product Calculation

When calculating the force on a charge moving in a magnetic field, understanding cross product calculations is key. The cross product, denoted by \(\mathbf{v} \times \mathbf{B}\), calculates a vector that is perpendicular to two given vectors \(\mathbf{v}\) (velocity) and \(\mathbf{B}\) (magnetic field).
This mathematical operation involves the use of a determinant to compute the cross product. Here's how it's calculated in practice:

• First, set up a 3x3 matrix, where the first row consists of the unit vectors \(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}\).
• The second row includes the components of the first vector, in this instance, components of \(\mathbf{v}\).
• The third row consists of the components of the second vector \(\mathbf{B}\).

Computing the determinant of this matrix provides the cross product result. For our example, it is calculated as follows:\[\mathbf{v} \times \mathbf{B} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ 3 & 4 & 1 \ 1 & 1 & -3 \end{vmatrix}\]This operation yields \(-13\hat{\mathbf{i}} + 10\hat{\mathbf{j}} - \hat{\mathbf{k}}\). This vector is perpendicular to both \(\mathbf{v}\) and \(\mathbf{B}\) and important for calculating the magnetic part of the Lorentz force. This cross product is essential in determining how magnetic fields influence moving charges.

Vector Mathematics

Vector mathematics is an essential aspect of physics and engineering, notably in the study of forces, fields, and movements. Vectors are quantities that have both magnitude and direction, represented typically in terms of components and unit vectors, such as \(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}\) in three-dimensional space.
Key aspects of vector mathematics include:

• Addition and subtraction: Vectors can be added or subtracted component-wise, aligning similar components together (e.g., \(\mathbf{A} = 3\hat{\mathbf{i}} + 4\hat{\mathbf{j}}\) and \(\mathbf{B} = \hat{\mathbf{i}} + \hat{\mathbf{j}}\) result in \(\mathbf{C} = (3+1)\hat{\mathbf{i}} + (4+1)\hat{\mathbf{j}}\)).
• Dot product: Used to find the angle or the projection between vectors, calculated as \(\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos\theta\).
• Cross product: As explained previously, the cross product results in a vector perpendicular to the plane formed by two vectors and is crucial when studying magnetic forces.

In the context of the Lorentz force, vector mathematics allows us to break down and understand the complex interactions in an electromagnetic field through clear steps and processes. It provides the tools to compute, not just the direction and magnitude of the resulting force, but also how different components of the field affect the force experienced by a moving charge.