Electrostatic Field

The electrostatic field is a field of force created by electric charges. It is a vector field that exerts a force on other electric charges placed within the field. The strength and direction of the electrostatic field depend on the amount of charge and the distance from the charge.

What is an Electrostatic Field?

An electrostatic field is the region around a charged particle where other charged particles experience an electrostatic force. The electric field \( \vec{E} \) at any point in space represents the force per unit charge that would be exerted on a test charge placed at that point.

Mathematical Definition of Electric Field

The electric field \( \vec{E} \) due to a point charge \( Q \) is defined as the force \( \vec{F} \) per unit positive test charge \( q \) placed at a distance \( r \) from the charge \( Q \). Mathematically, this is expressed as:

\[ \vec{E} = \frac{\vec{F}}{q} \]

Where:

\( \vec{F} \) is the force exerted on a test charge \( q \).
\( q \) is the test charge placed in the field.
\( \vec{E} \) is the electric field at the position of the test charge.

Coulomb’s Law

Coulomb’s Law gives the magnitude of the force between two point charges. The force \( \vec{F} \) between two point charges \( Q_1 \) and \( Q_2 \), separated by a distance \( r \), is given by:

\[ \vec{F} = k_e \frac{Q_1 Q_2}{r^2} \hat{r} \]

Where:

\( k_e \) is Coulomb's constant, \( k_e = 8.99 \times 10^9 \, \text{N m}^2 \text{C}^{-2} \).
\( Q_1 \) and \( Q_2 \) are the magnitudes of the charges.
\( r \) is the distance between the charges.
\( \hat{r} \) is the unit vector pointing from \( Q_1 \) to \( Q_2 \) (indicating the direction of the force).

Coulomb’s Law states that the force between two charges is attractive if the charges are of opposite sign (opposite charges attract), and repulsive if the charges are of the same sign (like charges repel). The force magnitude decreases with the square of the distance between the charges.

Electric Field of a Point Charge

The electric field \( \vec{E} \) created by a point charge \( Q \) at a distance \( r \) is given by:

\[ \vec{E} = k_e \frac{Q}{r^2} \hat{r} \]

Where:

\( Q \) is the point charge creating the electric field.
\( r \) is the distance from the charge to the point where the electric field is being calculated.
\( \hat{r} \) is the unit vector pointing away from the charge if \( Q \) is positive, or towards the charge if \( Q \) is negative.

The electric field \( \vec{E} \) points away from a positive charge and toward a negative charge. The magnitude of the electric field decreases with the square of the distance from the charge, as indicated by the \( r^2 \) term in the denominator.

Example: Electric Field Due to a Positive Point Charge

Let’s calculate the electric field created by a point charge \( Q = 5 \, \mu C \) at a distance of \( r = 0.2 \, \text{m} \) from the charge.

The magnitude of the electric field is given by:

\[ \vec{E} = k_e \frac{Q}{r^2} \]

Substituting values:

\[ \vec{E} = \left( 8.99 \times 10^9 \, \text{N m}^2 \text{C}^{-2} \right) \frac{5 \times 10^{-6} \, \text{C}}{(0.2 \, \text{m})^2} \]

\[ \vec{E} = 1.12 \times 10^5 \, \text{N/C} \]

Therefore, the electric field at a distance of 0.2 meters from a charge of \( 5 \, \mu C \) is \( 1.12 \times 10^5 \, \text{N/C} \), directed radially outward from the charge.

Superposition Principle for Electric Fields

The electric field due to a distribution of charges can be found by summing the electric fields due to individual charges. This is known as the superposition principle. If we have multiple point charges, the net electric field \( \vec{E}_{\text{net}} \) at a point is the vector sum of the electric fields \( \vec{E}_i \) due to each charge \( Q_i \):

\[ \vec{E}_{\text{net}} = \sum_{i} \vec{E}_i \]

Electric Field of a Continuous Charge Distribution

In many situations, the charge distribution is continuous (such as a uniformly charged rod or a spherical charge distribution). In this case, we compute the electric field by integrating over the distribution of charge. For example, for a linear charge distribution \( \lambda(x) \) along a rod, the electric field at a point is calculated by integrating the contributions from each infinitesimal charge element along the rod:

\[ \vec{E} = \frac{1}{4\pi \epsilon_0} \int \frac{\lambda(x) \, \hat{r}}{r^2} dx \]

Where:

\( \lambda(x) \) is the linear charge density.
\( \hat{r} \) is the unit vector pointing from the charge element to the point where the field is being calculated.
\( r \) is the distance from the charge element to the point of interest.

Applications of Electrostatic Fields

Electrostatic fields have numerous applications in various fields of science and engineering, including:

Capacitors: Devices that store electrical energy in an electrostatic field.
Electrostatic Precipitators: Used to remove particles from gases in industrial processes.
Electric Field Sensors: Instruments that detect the presence of electric fields in various environments.
Particle Accelerators: Devices that use electrostatic fields to accelerate charged particles to high velocities.

Conclusion

Understanding electrostatic fields is essential in physics and has widespread applications in modern technology and industry.