1 Answer
An electric field can be thought of as the force per unit charge experienced by a charged particle in the field. The relationship between electric field and area is best understood in the context of Gauss's law, which relates the electric field to the distribution of charge.
Here’s the explanation in simple terms:
\[
\Phi_E = \int \mathbf{E} \cdot dA = \frac{Q_{\text{enc}}}{\epsilon_0}
\]
Where:
- \( \Phi_E \) is the electric flux,
- \( \mathbf{E} \) is the electric field,
- \( dA \) is an infinitesimal area element,
- \( Q_{\text{enc}} \) is the charge enclosed within the surface, and
- \( \epsilon_0 \) is the permittivity of free space.
This is essentially saying that if you spread the same total charge over a larger area, the electric field at any given point (on that surface) becomes weaker.
\[
\Phi_E = E \cdot A
\]
Where:
- \( E \) is the electric field,
- \( A \) is the area.
For a fixed flux, if the area \( A \) increases, the electric field \( E \) must decrease to keep the product \( E \cdot A \) constant. Hence, the electric field is inversely proportional to the area:
\[
E \propto \frac{1}{A}
\]
In simple terms: If the area increases, the electric field strength decreases (for a fixed amount of charge). This is why the electric field is inversely proportional to the area of the cross-section.
Here’s the explanation in simple terms:
- Gauss’s Law: It states that the electric flux (the total amount of electric field passing through a surface) is proportional to the charge enclosed within that surface. Mathematically, it is written as:
\[
\Phi_E = \int \mathbf{E} \cdot dA = \frac{Q_{\text{enc}}}{\epsilon_0}
\]
Where:
- \( \Phi_E \) is the electric flux,
- \( \mathbf{E} \) is the electric field,
- \( dA \) is an infinitesimal area element,
- \( Q_{\text{enc}} \) is the charge enclosed within the surface, and
- \( \epsilon_0 \) is the permittivity of free space.
- Electric Field and Area: Now, imagine you have a surface with a given area. The electric flux through that surface depends on both the strength of the electric field and the size of the area. The electric field is inversely proportional to the area when we consider the total electric flux through a surface. This is because, for a constant charge enclosed within the surface, as the area increases, the electric field at any point on the surface has to decrease to maintain the same amount of flux.
This is essentially saying that if you spread the same total charge over a larger area, the electric field at any given point (on that surface) becomes weaker.
- Inversely Proportional: Mathematically, for a uniform electric field, the electric flux \( \Phi_E \) is given by:
\[
\Phi_E = E \cdot A
\]
Where:
- \( E \) is the electric field,
- \( A \) is the area.
For a fixed flux, if the area \( A \) increases, the electric field \( E \) must decrease to keep the product \( E \cdot A \) constant. Hence, the electric field is inversely proportional to the area:
\[
E \propto \frac{1}{A}
\]
In simple terms: If the area increases, the electric field strength decreases (for a fixed amount of charge). This is why the electric field is inversely proportional to the area of the cross-section.