Electric Flux Formula

Electric Flux Formula

Electric flux is a fundamental concept in electromagnetism that quantifies the flow of electric field through a given surface. It represents the total number of electric field lines passing through a specified area, taking into account both the strength and direction of the electric field. Electric flux is determined by Gauss’s Law, which relates the electric flux to the total electric charge enclosed by the surface and the permittivity of the medium. Electric flux plays a crucial role in understanding the behavior of electric fields, their interactions with charged objects, and various practical applications in engineering and physics. Learn more about electric flux in this article by Extramarks.

What is Electric Flux?

Electric flux is a fundamental concept in electromagnetism that quantifies the flow of electric field through a given surface. It represents the total number of electric field lines passing through a specified area, taking into account both the strength and direction of the electric field. Essentially, electric flux measures how much electric field penetrates or intersects a surface.

If there is no net charge within a closed surface, every field line put into it travels into the interior and is often directed outward somewhere on the surface. The negative flux is equal in size to the positive flux, resulting in a net or total electric flux is equal to zero.

If a net charge is confined within a closed surface, the total flux through the surface is proportional to the enclosed charge, which is positive if positive and negative otherwise.

Formula of Electric Flux

The mathematical relationship between electric flux and the contained charge is known as Gauss’ law of the electric field.

The formula for electric flux (\( \Phi_E \)) through a surface is given by:

\[ \Phi_E = \vec{E} \cdot \vec{A} \]

Where:
\( \vec{E} \) is the electric field vector,
\( \vec{A} \) is the area vector of the surface, and
\( \cdot \) denotes the dot product.

In component form, if \( \vec{E} = E_x\hat{i} + E_y\hat{j} + E_z\hat{k} \) and \( \vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k} \), then the dot product \( \vec{E} \cdot \vec{A} \) is calculated as:

\[ \vec{E} \cdot \vec{A} = E_x A_x + E_y A_y + E_z A_z \]

The resulting electric flux (\( \Phi_E \)) is a scalar quantity, typically measured in units of volts per meter (V.m). It represents the total electric field passing through the surface, taking into account both the magnitude and direction of the electric field and the orientation of the surface.

Thus, In the same way that the plane is normal to the flow of the electric field in the preceding example, the total flux is given as:

Electric Flux Formula  = EA cos θ

Where,

• E is the magnitude of the electric field
• A is the area of the surface through which the electric flux is to be calculated
• θ is the angle made by the plane and the axis parallel to the direction of flow of the electric field

Unit and Dimension of Electric Flux

The SI unit of Electric Flux is Volt-meter or Vm. The other unit for measurement of electric flux is Nm²C^-1

The dimension of electric flux is ML³T⁻³A⁻¹

Properties of Electric Flux

Electric flux possesses several properties that are essential for understanding its behavior and applications in electromagnetism. Here are some key properties:

1. Scalar Quantity: Electric flux (\( \Phi_E \)) is a scalar quantity. This means it has magnitude but no direction. It represents the total electric field passing through a surface, regardless of the orientation of the surface.

2. Dependence on Electric Field and Surface Area: Electric flux is directly proportional to the magnitude of the electric field (\( \vec{E} \)) and the area (\( \vec{A} \)) of the surface. As the electric field strength or area of the surface increases, the electric flux through the surface also increases.

3. Dependence on Surface Orientation: The orientation of the surface with respect to the electric field affects the electric flux. When the surface is perpendicular to the electric field lines, the electric flux is maximum. Conversely, when the surface is parallel to the electric field lines, the electric flux is zero.

4. Gauss’s Law: Electric flux is governed by Gauss’s Law, a fundamental principle in electromagnetism. Gauss’s Law relates the total electric flux through a closed surface to the total charge enclosed by the surface and the permittivity of the medium. It provides a powerful tool for calculating electric flux in various situations.

5. Conservation Principle: In a closed system, the total electric flux entering a region is equal to the total electric flux leaving the region. This conservation principle stems from the fact that electric field lines are continuous and cannot start or end at arbitrary points in space.

6. Relation to Electric Potential: Electric flux is related to electric potential through the gradient theorem. The electric flux through a surface is equal to the negative of the rate of change of electric potential with respect to distance along the surface normal.

Applications of Electric Flux

Electric flux, a fundamental concept in electromagnetism, finds applications in various fields, ranging from physics and engineering to everyday technologies. Here are some key applications:

1. Gauss’s Law and Charge Distributions: It is extensively used in physics and engineering to calculate electric fields around charged objects and understand the behavior of electrically charged systems.

2. Electric Field Mapping: Electric flux is utilized in mapping electric fields around charged objects or systems.

3. Electrostatic Shielding: Electric flux plays a crucial role in electrostatic shielding, a technique used to protect sensitive electronic devices from external electric fields. By surrounding the device with conductive materials, electric flux is diverted away from the interior, minimizing interference and ensuring the proper functioning of the device.

4. Capacitors: Electric flux is fundamental to the operation of capacitors, which store electric charge and energy. In capacitors, electric flux is used to calculate capacitance, the ability of a capacitor to store charge per unit voltage. Understanding electric flux is essential for designing and optimizing capacitor systems for various applications, such as energy storage and signal processing.

5. Electrostatic Precipitators: Electric flux is employed in electrostatic precipitators, devices used to remove particulate matter (such as dust and smoke) from industrial exhaust gases. By applying high-voltage electric fields, electric flux is used to impart electric charges to particles, causing them to be attracted to collection plates and removed from the gas stream.

6. Electrostatic Painting: Electric flux is utilized in electrostatic painting, a process used to apply paint coatings to various surfaces. By charging paint particles and applying high-voltage electric fields, electric flux is used to attract paint particles to the surface being painted, resulting in uniform and efficient coating deposition.

Solved Examples using Electric Flux Formula

Example 1: An electric field of 800 V/m makes an angle of 30 degree with the surface vector. It has a magnitude of 0.800 m2. Find the electric flux that passes through the surface.

Solution:

The electric flux which is passing through the surface is given by the equation as:

Φ_E = E.A = EA cos θ

Φ_E = (800 V/m) (0.800 m²) cos 30

Φ_E = 320 V m

Example 2: Calculate the electric flux through a rectangular surface of dimensions \( 4 \, \text{m} \times 3 \, \text{m} \), placed in a uniform electric field of magnitude \( 100 \, \text{N/C} \), with the electric field lines perpendicular to the surface.

Solution:

Given:
Area of the rectangular surface (\( A \)) = \( 4 \, \text{m} \times 3 \, \text{m} = 12 \, \text{m}^2 \)
Electric field magnitude (\( E \)) = \( 100 \, \text{N/C} \)
Angle between electric field lines and surface (\( \theta \)) = \( 0^\circ \) (since the electric field lines are perpendicular to the surface)

Using the formula for electric flux:
\[ \Phi_E = \vec{E} \cdot \vec{A} \]

\[ \Phi_E = E \cdot A \cdot \cos(\theta) \]

Since \( \theta = 0^\circ \) and \( \cos(0^\circ) = 1 \), we have:
\[ \Phi_E = E \cdot A \cdot 1 \]

Substituting the given values:
\[ \Phi_E = (100 \, \text{N/C}) \times (12 \, \text{m}^2) \]
\[ \Phi_E = 1200 \, \text{N}\cdot\text{m}^2/\text{C} \]

Therefore, the electric flux through the rectangular surface is \( 1200 \, \text{N}\cdot\text{m}^2/\text{C} \).

Example 3: A spherical shell of radius \( 2 \, \text{m} \) has a uniform electric field of magnitude \( 500 \, \text{N/C} \) passing through it. Calculate the electric flux through the spherical shell.

Solution:

Given:
Radius of the spherical shell (\( r \)) = \( 2 \, \text{m} \)
Electric field magnitude (\( E \)) = \( 500 \, \text{N/C} \)

The surface area of a sphere is given by \( A = 4\pi r^2 \).
\[ A = 4\pi (2 \, \text{m})^2 \]
\[ A = 4\pi (4 \, \text{m}^2) \]
\[ A = 16\pi \, \text{m}^2 \]

Using the formula for electric flux:
\[ \Phi_E = \vec{E} \cdot \vec{A} \]
\[ \Phi_E = E \cdot A \cdot \cos(\theta) \]

Since the electric field lines are radially outward and the normal vector of the spherical surface is also radially outward, \( \theta = 0^\circ \). Therefore, \( \cos(\theta) = 1 \).

\[ \Phi_E = E \cdot A \cdot 1 \]

Substituting the given values:
\[ \Phi_E = (500 \, \text{N/C}) \times (16\pi \, \text{m}^2) \]
\[ \Phi_E = 8000\pi \, \text{N}\cdot\text{m}^2/\text{C} \]

Therefore, the electric flux through the spherical shell is \( 8000\pi \, \text{N}\cdot\text{m}^2/\text{C} \).