Problem 47 - Arihant AIEEE Physics

Question

In a region, electric field varies as \(E=2 x^{2}-4\) where \(x\) is distance in S.I. from origin along \(x\) -axis. A positive charge of \(1 \mu \mathrm{C}\) is released with minimum velocity from infinity for crossing the origin, then: (a) the kinetic energy at the origin must be zero (b) the kinetic energy at the origin may be zero (c) the kinetic energy at \(x=\sqrt{2} \mathrm{~m}\) must be zero (d) the kinetic energy at \(x=2 \mathrm{~m}\) mav be zero

Step-by-Step Solution

Answer

(a) and (d) are correct.

Step 1: Understanding the Electric Field

The electric field is given as a function of the distance, \(E = 2x^2 - 4\). This means the electric field depends on the square of the distance from the origin along the \(x\)-axis.

Step 2: Concept of Work and Electric Potential

The work done by the electric field on a charge is defined by the change in electric potential energy, \(U\). The change in potential energy is equivalently the line integral of the electric field, \(U = - \int E \, dx\). If a particle travels from infinity to a point, its kinetic energy change equals the work done on it.

Step 3: Integrate Electric Field to Find Potential

Calculate the electrical potential function by integrating the electric field from infinity to a point \(x\).\[ V(x) = - \int E \, dx = - \int (2x^2 - 4) \, dx \]Carrying out the integration:\[ V(x) = - \left( \frac{2}{3}x^3 - 4x \right) = - \frac{2}{3}x^3 + 4x + C \]

Step 4: Determine Constant of Integration

As the charge is released from infinity, we're considering the potential at infinity to be zero, setting the constant \(C = 0\). So,\[ V(x) = -\frac{2}{3}x^3 + 4x \]

Step 5: Calculate Kinetic Energy at Specific Points

Kinetic energy \(K\) at the origin implies total energy change from infinity to the origin is zero, thus kinetic energy should offset the potential energy.\[ K = q(V_{\infty} - V(0)) = 0 \] Since \(V_{\infty} = 0\), the potential at the origin is zero, which implies kinetic energy is zero at X=\(\sqrt{2}\). Partial derivatives calculate the specific condition.

Step 6: Evaluate the Options

- (a) is true since kinetic energy at \(x=0\) is zero by conservation of energy.- (b) must be false given potential zero condition needed for zero kinetic energy.- (c) is false since electric potential at \(x=\sqrt{2}\) is non-zero making kinetic energy non-zero as well. - (d) is correct because potential results small.

Key Concepts

Electromagnetism

Electromagnetism is a branch of physics that deals with the study of electric fields, magnetic fields, and how they interact with matter. An electric field is a region around a charged particle where a force would be experienced by other charges. In this exercise, the electric field is given by the equation:
\( E = 2x^2 - 4 \), where \(x\) is the distance from the origin along the x-axis. This indicates that the electric field varies with the square of the distance from the origin. Thus, as you move along the x-axis, the magnitude and direction of the field change.
Understanding the behavior of an electric field is crucial for predicting how charged particles will move. It's like a guide that helps us determine the path a charged object will take when placed in this field. Knowing this can be extremely useful in fields like electronics and electromagnetic theory, where control of particle movement is essential.

Kinetic Energy

Kinetic Energy is the energy an object possesses due to its motion. It is mathematically represented as:
\( K = \frac{1}{2}mv^2 \), where \(m\) is the mass and \(v\) is the velocity of the object.
In this exercise, a positive charge of \(1 \mu \text{C}\) is released from infinity and eventually crosses the origin. As it travels towards the origin, it converts potential energy into kinetic energy.

• At the origin, the kinetic energy is calculated by considering the total change in energy from infinity to the origin.
• The principle of energy conservation dictates that the change in kinetic energy must equal the change in electric potential energy as it moves through the electric field.
• If the potential energy at the origin is zero (as is the case in this problem), this implies that the kinetic energy is also zero when the particle is at the origin.

This balance of energy ensures that the particle behaves predictably within the electric field, following the conservation of energy principle, which is a fundamental tenet of physics.

Electric Potential

Electric potential is a measure of the potential energy per unit charge at a point in a field. It's essentially how much energy would be required or liberated to move a charge from one point to another within an electric field. In mathematical terms, we find it by integrating the electric field. In the provided formula:
\[ V(x) = - \int E \, dx = - \int (2x^2 - 4) \, dx \]
This results in
\[ V(x) = -\frac{2}{3}x^3 + 4x + C \]

• The constant \(C\) is determined based on the boundary conditions, such as setting the potential at an infinite distance to zero.
• In this scenario, the charge needs to cross the origin, thus understanding how electric potential varies along its path helps predict points of zero or maximum kinetic energy.
• At point \(x=\sqrt{2}\), despite the electric field, the potential energy is not zero, indicating the kinetic energy will also not be zero at that point.
• Electric potential directly influences the motion and energy conversion of charges in a field, helping us grasp how forces act at various locations in the space around a charge.