The electric field intensity (E) at point r due to a point charge (Q) located at r’ is: 

Explanation:

Electric Field Intensity Due to a Point Charge

Definition: The electric field intensity (E) at a point in space due to a point charge (Q) is a vector quantity that represents the force experienced by a unit positive charge placed at that point. It is given by Coulomb's law, which describes the magnitude and direction of the electric field created by a point charge.

Mathematical Expression: The electric field intensity (E) at point r due to a point charge (Q) located at r' is given by the formula:

Correct Option: Option 2

\(\rm E=\frac{Q(r-r')}{4\pi \varepsilon_0(|r-r'|^3)}\)

This formula correctly represents the electric field intensity created by a point charge. The vector \( \rm r-r' \) points from the charge location to the point where the field is being calculated, and the magnitude is the square of the distance between these points. The denominator \( (|r-r'|^3) \) accounts for the inverse square law, while the direction is maintained by the vector form.

Explanation:

To understand why this is the correct formula, let's break down the components:

• Q: The magnitude of the point charge creating the electric field.
• r: The position vector of the point where the electric field is being calculated.
• r': The position vector of the point charge.
• ε₀: The permittivity of free space, a constant that characterizes the strength of the electric field in a vacuum.
• r - r': The vector pointing from the charge location to the point of interest.
• |r - r'|: The magnitude of the vector distance between the charge location and the point of interest.

The formula follows Coulomb's law, which states that the electric field due to a point charge decreases with the square of the distance from the charge and acts along the line connecting the charge and the point of interest.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\rm E=\frac{Q(r'-r)}{4\pi \varepsilon_0(|r'-r|^2)}\)

This option is incorrect because it has the wrong vector direction and the incorrect power of the distance in the denominator. The correct expression should have the vector \( r - r' \) and the distance raised to the power of three in the denominator.

Option 3: \(\rm E=\frac{Q(r'-r)}{4\pi \varepsilon_0(|r'-r|^3)}\)

This option is incorrect because it has the wrong vector direction. The vector should point from the charge to the point where the field is being calculated, so it should be \( r - r' \).

Option 4: \(\rm E=\frac{Q(r-r')}{4\pi \varepsilon_0(|r-r'|^2)}\)

This option is incorrect because it has the correct vector direction but the incorrect power of the distance in the denominator. The electric field decreases with the square of the distance, but the vector form of the equation requires the magnitude to be cubed.

Conclusion:

Understanding the correct expression for the electric field intensity due to a point charge is essential for solving problems in electrostatics. The correct formula, \(\rm E=\frac{Q(r-r')}{4\pi \varepsilon_0(|r-r'|^3)}\), correctly incorporates the direction and magnitude of the electric field according to Coulomb's law. This formula is fundamental in various applications, including calculating forces between charges and understanding electric field distributions in different configurations.