Electric field
Understanding the Electric Field
The electric field is a fundamental concept in electrostatics, which describes the influence that a charged object exerts on other charges in the space around it. It is a vector field, meaning it has both magnitude and direction.
Definition
The electric field E at a point in space is defined as the force F experienced by a small positive test charge q placed at that point, divided by the magnitude of the test charge:
$$ \mathbf{E} = \frac{\mathbf{F}}{q} $$
The units of electric field are newtons per coulomb (N/C).
Properties of Electric Fields
- Vector Quantity: The electric field has both magnitude and direction.
- Superposition Principle: The net electric field due to multiple charges is the vector sum of the individual fields created by each charge.
- Lines of Force: Electric field lines start on positive charges and end on negative charges, never crossing each other.
Electric Field Due to a Point Charge
The electric field due to a point charge Q at a distance r from the charge is given by Coulomb's law:
$$ \mathbf{E} = k \frac{|Q|}{r^2} \hat{r} $$
where:
- k is Coulomb's constant ((8.9875 \times 10^9 \, \text{N m}^2/\text{C}^2)),
- Q is the charge creating the field,
- r is the distance from the charge to the point of interest,
- (\hat{r}) is the unit vector pointing from the charge to the point of interest.
Electric Field of a Continuous Charge Distribution
For a continuous charge distribution, the electric field is calculated by integrating the contributions from each infinitesimal charge element:
$$ \mathbf{E} = \int d\mathbf{E} = \int k \frac{dq}{r^2} \hat{r} $$
where dq is an infinitesimal charge element of the distribution.
Examples
Example 1: Electric Field of a Single Point Charge
Calculate the electric field 0.1 m away from a point charge of (2 \times 10^{-6}) C.
$$ \mathbf{E} = k \frac{|Q|}{r^2} = (8.9875 \times 10^9) \frac{2 \times 10^{-6}}{(0.1)^2} = 1.7975 \times 10^6 \, \text{N/C} $$
Example 2: Superposition of Electric Fields
Calculate the net electric field at a point P due to two charges, (Q_1 = 5 \times 10^{-6}) C located at (0, 0) and (Q_2 = -3 \times 10^{-6}) C located at (0, 2) meters.
First, calculate the electric field due to each charge at point P, then add the vector fields to find the net field.
Differences and Important Points
| Property | Electric Field | Gravitational Field |
|---|---|---|
| Source | Charges | Masses |
| Force Direction | Radial, away from positive and towards negative charges | Radial, towards the mass |
| Field Lines | Start on positive and end on negative charges | Always point towards the mass |
| Formula | ( \mathbf{E} = k \frac{ | Q |
| Superposition | Yes | Yes |
| Relative Strength | Much stronger than gravitational force | Weaker than electric force |
Conclusion
The electric field is a crucial concept in understanding how charges interact with each other. It provides a way to calculate the force on a charge without having to consider the direct interaction with other charges. By understanding electric fields, one can predict the behavior of charges in various configurations and under different conditions.