6.3 Magnetic Flux
A plane of surface area A placed in a uniform magnetic field B.
Faraday’s great insight lay in discovering a simple mathematical relation to explain the series of experiments he carried out on electromagnetic induction. However, before we state and appreciate his laws, we must get familiar with the notion of magnetic flux, \(\Phi_{\mathrm{B}}\) . Magnetic flux is defined in the same way as electric flux is defined in Chapter 1. Magnetic flux through a plane of area A placed in a uniform magnetic field B (Fig. 6.4) can be written as
\[ \Phi_{\mathrm{B}}=\mathbf{B} \cdot \mathbf{A}=B A \cos \theta \]
where \(\theta \) is angle between B and A. The notion of the area as a vector has been discussed earlier in Chapter 1. Equation (6.1) can be extended to curved surfaces and nonuniform fields.
Magnetic field B i at the i th area element. dA i represents area vector of the i th area element.
If the magnetic field has different magnitudes and directions at various parts of a surface as shown in Fig. 6.5, then the magnetic flux through the surface is given by
\[ \Phi_{\mathrm{B}}=\mathbf{B}_{1} \cdot \mathrm{d} \mathbf{A}_{1}+\mathbf{B}_{2} \cdot \mathrm{d} \mathbf{A}_{2}+\cdots=\sum_{\mathrm{sill}} \mathbf{B}_{1} \cdot \mathrm{d} \mathbf{A}_{\mathrm{i}} \]
where ‘all’ stands for summation over all the area elements \( \mathrm{dA}_{i} \) comprising the surface and \( \mathbf{B}_{i} \) is the magnetic field at the area element \( \mathrm{dA}_{i} \) . The SI unit of magnetic flux is weber (Wb) or tesla meter squared ( \(Tm^{2}\) ). Magnetic flux is a scalar quantity.