Magnetic Flux Formula

Magnetic Flux Formula

Magnetic flux is a fundamental concept in physics that describes the quantity of magnetic field passing through a given area. It is a measure of how much magnetic field penetrates a surface or area that is oriented perpendicular to the direction of the magnetic field. The magnetic flux formula is given as Φ_B=B⋅Acos(θ).

Magnetic flux finds applications across various disciplines, including electromagnetism, electrical engineering, and astrophysics. Learn more about magnetic flux, its definition, formula and examples

What is Magnetic?

Magnetic flux is a fundamental concept in electromagnetism that describes the quantity of magnetic field passing through a given area. It is analogous to the concept of electric flux in electrostatics but specifically applies to magnetic fields.

Magnetic Flux Definition

Magnetic flux through a surface is defined as the total magnetic field passing through that surface. Mathematically, it is given by the dot product of the magnetic field vector and the area vector perpendicular to the surface.

Magnetic Flux Symbol

The symbol of magnetic flux is Φ_B

Magnetic Flux Formula

The formula for magnetic flux (\(\Phi_B\)) is derived from the dot product of the magnetic field vector (\(\mathbf{B}\)) and the area vector (\(\mathbf{A}\)) through which the magnetic field lines pass. Here’s how it is expressed:

\[ \Phi_B = \mathbf{B} \cdot \mathbf{A} \]

In scalar form, this can be written as:

\[ \Phi_B = B \cdot A \cdot \cos(\theta) \]

where:
\(\Phi_B\) is the magnetic flux,
\(\mathbf{B}\) is the magnetic field vector (measured in tesla, T),
\(\mathbf{A}\) is the area vector perpendicular to the magnetic field (measured in square meters, m\(^2\)),
\(B\) is the magnitude of the magnetic field (\(|\mathbf{B}|\)),
\(A\) is the magnitude of the area (\(|\mathbf{A}|\)),
\(\theta\) is the angle between \(\mathbf{B}\) and \(\mathbf{A}\).

Unit of Magnetic Flux

The SI unit of magnetic flux (\(\Phi_B\)) is the weber (Wb), which is equivalent to:

\[ 1 \, \text{Wb} = 1 \, \text{T} \cdot \text{m}^2 \]

Direction of \(\mathbf{A}\): The area vector \(\mathbf{A}\) must be perpendicular to the magnetic field \(\mathbf{B}\) to accurately calculate magnetic flux.

Cosine of Angle: The cosine term \(\cos(\theta)\) accounts for the orientation of the area vector relative to the magnetic field. It ensures that only the component of the magnetic field perpendicular to the area contributes to the flux calculation.

Faraday’s Law of Electromagnetic Induction

According to Faraday’s Law of Electromagnetic Induction, a changing magnetic flux through a circuit induces an electromotive force (emf) and thus an electric current in the conductor.

This principle is fundamental to the operation of electric generators, transformers, and various electromechanical devices.

Applications of Magnetic Flux

• Magnetic Sensors: Sensors such as Hall effect sensors and magnetometers measure magnetic flux density (or magnetic field strength) to detect magnetic fields, position, and motion. These sensors find applications in automotive, aerospace, and consumer electronics industries.
• Magnetic Recording: Magnetic flux is crucial in magnetic storage devices like hard drives and magnetic tapes. Data is stored and retrieved by changing the magnetic flux patterns on recording media.
• Electric Motors and Generators: Electric motors convert electrical energy into mechanical energy through the interaction of magnetic fields and conductors. The principles of magnetic flux are crucial for optimizing motor efficiency and performance.
• Power Transformers: Power transformers regulate voltage levels and transmit electrical power efficiently across power grids. Magnetic flux calculations ensure proper coupling between primary and secondary windings to minimize losses.
• MRI (Magnetic Resonance Imaging): MRI scanners use strong magnetic fields and varying magnetic flux to create detailed images of internal body structures. The interaction between magnetic flux and hydrogen atoms in tissues provides diagnostic information without the use of ionizing radiation.

Solved Examples on Magnetic Flux Formula

Example 1: Calculate the magnetic flux through a rectangular surface of dimensions \( 0.1 \, \text{m} \times 0.2 \, \text{m} \), placed in a uniform magnetic field of \( 0.5 \, \text{T} \). The angle between the magnetic field and the normal to the surface is \( 30^\circ \).

Solution:

Given:
Dimensions of the rectangular surface: \( 0.1 \, \text{m} \times 0.2 \, \text{m} \)
Magnetic field (\(\mathbf{B}\)): \( 0.5 \, \text{T} \)
Angle (\(\theta\)) between \(\mathbf{B}\) and the normal to the surface: \( 30^\circ \)

Calculate the area (\(A\)) of the surface:
\[ A = 0.1 \, \text{m} \times 0.2 \, \text{m} = 0.02 \, \text{m}^2 \]

Convert the angle to radians (if necessary):
\[ \theta = 30^\circ = \frac{\pi}{6} \, \text{radians} \]

Calculate the magnetic flux (\(\Phi_B\)):
\[ \Phi_B = B \cdot A \cdot \cos(\theta) \]
\[ \Phi_B = 0.5 \, \text{T} \cdot 0.02 \, \text{m}^2 \cdot \cos\left(30^\circ\right) \]
\[ \Phi_B = 0.5 \cdot 0.02 \cdot \cos\left(\frac{\pi}{6}\right) \]
\[ \Phi_B = 0.01 \cdot \cos\left(\frac{\pi}{6}\right) \]
\[ \Phi_B = 0.01 \cdot \frac{\sqrt{3}}{2} \]
\[ \Phi_B = 0.01 \cdot 0.866 \]
\[ \Phi_B = 0.00866 \, \text{Wb} \]

Result: The magnetic flux through the rectangular surface is \( 0.00866 \, \text{Wb} \).

Example 2: A circular coil with a radius of \( 0.1 \, \text{m} \) is placed in a magnetic field of \( 0.3 \, \text{T} \). If the coil is rotated from an initial position where the magnetic flux through it is zero to a final position where the magnetic field lines are perpendicular to the coil, calculate the change in magnetic flux.

Solution:

Given:
Radius of the circular coil (\( r \)): \( 0.1 \, \text{m} \)
Magnetic field (\(\mathbf{B}\)): \( 0.3 \, \text{T} \)

Initial Magnetic Flux (\(\Phi_{B1}\)) when the coil is perpendicular to the magnetic field:
\[ \Phi_{B1} = B \cdot A \cdot \cos(0^\circ) \]
\[ \Phi_{B1} = 0.3 \, \text{T} \cdot (\pi \cdot 0.1^2 \, \text{m}^2) \cdot 1 \]
\[ \Phi_{B1} = 0.3 \cdot \pi \cdot 0.01 \]
\[ \Phi_{B1} = 0.00942 \, \text{Wb} \]

Final Magnetic Flux (\(\Phi_{B2}\)) when the coil is perpendicular to the magnetic field:
\[ \Phi_{B2} = B \cdot A \cdot \cos(90^\circ) \]
\[ \Phi_{B2} = 0.3 \, \text{T} \cdot (\pi \cdot 0.1^2 \, \text{m}^2) \cdot 0 \]
\[ \Phi_{B2} = 0 \, \text{Wb} \]

Change in Magnetic Flux (\(\Delta \Phi_B\)):
\[ \Delta \Phi_B = \Phi_{B2} – \Phi_{B1} \]
\[ \Delta \Phi_B = 0 \, \text{Wb} – 0.00942 \, \text{Wb} \]
\[ \Delta \Phi_B = -0.00942 \, \text{Wb} \]

Result: The change in magnetic flux as the coil rotates from initial to final position is \( -0.00942 \, \text{Wb} \).

Example 3: A circular loop of radius \( 0.05 \, \text{m} \) is placed in a uniform magnetic field of \( 0.4 \, \text{T} \). Calculate the magnetic flux through the circular loop when the magnetic field makes an angle of \( 60^\circ \) with the normal to the plane of the loop.

Solution:

Given:
Radius of the circular loop (\( r \)): \( 0.05 \, \text{m} \)
Magnetic field (\(\mathbf{B}\)): \( 0.4 \, \text{T} \)
Angle (\(\theta\)) between \(\mathbf{B}\) and the normal to the plane of the loop: \( 60^\circ \)

Area of the circular loop (\( A \)):
\[ A = \pi \cdot r^2 \]
\[ A = \pi \cdot (0.05 \, \text{m})^2 \]
\[ A = \pi \cdot 0.0025 \, \text{m}^2 \]
\[ A = 0.00785 \, \text{m}^2 \]

Calculate the magnetic flux (\(\Phi_B\)):
\[ \Phi_B = B \cdot A \cdot \cos(\theta) \]
\[ \Phi_B = 0.4 \, \text{T} \cdot 0.00785 \, \text{m}^2 \cdot \cos(60^\circ) \]
\[ \Phi_B = 0.4 \cdot 0.00785 \cdot 0.5 \]
\[ \Phi_B = 0.00157 \, \text{Wb} \]

Result: The magnetic flux through the circular loop is \( 0.00157 \, \text{Wb} \).