How to Derive the Dimensional Formula of Magnetic Flux
Magnetic flux is an important physical quantity in electromagnetism, describing the total magnetic field passing through a given surface. Understanding the dimensions of magnetic flux is essential for solving related problems in physics and engineering, especially in the context of competitive examinations like JEE Main.
Definition and Formula of Magnetic Flux
Magnetic flux, denoted by $\phi$, is defined as the scalar product of the magnetic field $\vec{B}$ and the perpendicular area $\vec{S}$ through which the field passes. The mathematical expression is $\phi = \vec{B} \cdot \vec{S}$.
Units of Magnetic Flux
The SI unit of magnetic flux is the weber (Wb), while the CGS unit is the maxwell (Mx). In SI, $1\,\mathrm{Wb} = 1\,\mathrm{T} \cdot \mathrm{m}^2$. The relation between SI and CGS units is $1\,\mathrm{Wb} = 10^8\,\mathrm{Mx}$.
| Unit System | Unit of Magnetic Flux |
|---|---|
| SI | Weber (Wb) |
| CGS | Maxwell (Mx) |
Dimensional Formula of Magnetic Flux
Magnetic flux is the product of magnetic field ($B$) and area ($S$). The dimensional formula of area is $[L^2]$. The dimensional formula of magnetic field ($B$) is $[M\,T^{-2}\,A^{-1}]$.
Therefore, the dimensional formula for magnetic flux is obtained as follows:
$\phi = B \times S$
$[B] = [M\,T^{-2}\,A^{-1}]$ and $[S] = [L^2]$
$[\phi] = [M\,T^{-2}\,A^{-1}][L^2] = [M\,L^2\,T^{-2}\,A^{-1}]$
Explanation of Each Dimension
In the dimensional formula $[M\,L^2\,T^{-2}\,A^{-1}]$, $M$ represents mass, $L$ denotes length, $T$ indicates time, and $A$ is electric current. These fundamental dimensions combine to characterize magnetic flux physically.
Conditions for Maximum and Minimum Magnetic Flux
Magnetic flux is maximum when the surface is perpendicular to the magnetic field ($\theta = 0^\circ$). It is minimum (zero) when the surface is parallel to the field ($\theta = 90^\circ$), since $\phi = B S \cos \theta$.
Comparison with Related Quantities
The dimensions of magnetic flux are distinct from those of magnetic flux density. To learn more about the dimensions of magnetic flux density and other electromagnetic quantities, refer to Dimensions Of Magnetic Flux.
Related Physical Quantities
The concept of magnetic flux is related to other quantities such as electric flux, resistance, permittivity, force, and viscosity. Their respective dimensional formulas can be explored for comparative understanding, as detailed in Dimensions Of Electric Flux and Dimensions Of Force.
FAQs on Understanding the Dimensions of Magnetic Flux
1. What are the dimensions of magnetic flux?
Magnetic flux has the dimensions of mass × length2 × time−2 × electric current−1, commonly written as [M L2 T−2 I−1].
In summary:
- Formula: Magnetic flux (Φ) = Magnetic field (B) × Area (A) × cosθ
- SI unit: Weber (Wb)
- Dimensional formula: [M L2 T−2 I−1]
2. What is magnetic flux and its SI unit?
Magnetic flux measures the total magnetic field passing through a given area.
Key points to remember:
- SI unit: Weber (Wb)
- Symbol: Φ or ΦB
- Formula: Φ = B × A × cosθ
3. How do you derive the dimensional formula of magnetic flux?
To derive the dimensional formula of magnetic flux, use the relation Φ = B × A:
- B (magnetic field) = [M T−2 I−1]
- A (area) = [L2]
Φ = [M L2 T−2 I−1]
4. What is the formula for magnetic flux?
The formula for magnetic flux is:
Φ = B × A × cosθ
- Φ = Magnetic flux
- B = Magnetic field strength
- A = Area vector
- θ = Angle between B and the normal to the area
5. What is the significance of the negative sign in Faraday's law of electromagnetic induction?
The negative sign in Faraday's law indicates the direction of the induced EMF, which opposes the change in magnetic flux as per Lenz's law.
- Significance: Ensures conservation of energy
- Shows opposition: The induced current’s direction always resists the flux change
6. What are the units and dimensions of magnetic field and magnetic flux?
Magnetic field (B):
- SI unit: Tesla (T)
- Dimensional formula: [M T−2 I−1]
- SI unit: Weber (Wb)
- Dimensional formula: [M L2 T−2 I−1]
7. Is magnetic flux a scalar or vector quantity?
Magnetic flux is a scalar quantity.
Important points:
- It has magnitude but no direction
- Depends on the orientation of surface and field
8. Define Weber. How is it related to magnetic flux?
Weber (Wb) is the SI unit of magnetic flux.
- 1 Weber = Magnetic flux through an area of 1 m2 with a magnetic field of 1 Tesla perpendicular to the surface
- Relation: 1 Wb = 1 T × 1 m2
9. What factors affect the magnitude of magnetic flux through a surface?
Magnetic flux through a surface depends on:
- Strength of magnetic field (B)
- Area of the surface (A)
- Angle (θ) between field and area vector
10. What is the physical meaning of magnetic flux?
The physical meaning of magnetic flux is the total number of magnetic field lines passing through a given area.
- More field lines = Higher magnetic flux
- Measures how much of the magnetic field "passes through" a surface
