Understanding Faraday’s and Ampere’s Laws in Electromagnetism
Faraday’s Law of Electromagnetic Induction
Faraday’s Law expresses the induced electromotive force (EMF) as a function of the variation of magnetic flux (ΦB):
EMF = – ΔΦB / Δt
The electromotive force induced in a coil is equal to the rate of change of the magnetic flux through the surface bounded by the loop. For a coil with N turns, the EMF induced is:
EMF = -N * (ΔΦB / Δt)
If the coil has tightly wound turns, the flux through each turn will be the same, and the EMF induced in the coil is the sum of the individual EMFs.
Ampere’s Law and Magnetic Fields
Magnetic Field of a Straight Wire
In 1820, Oersted discovered that electric current creates a magnetic field. Shortly after, Biot published research on the magnetic field created by a straight wire carrying current.
The magnetic field lines due to the wire are concentric circles around the wire. The direction of the field can be determined using the right-hand rule.
Considering a point P at a distance ‘r’ from a straight wire carrying a current ‘I’, the magnetic field vector B at that point is tangent to a circle centered on the wire. The magnitude of the magnetic field is directly proportional to the current ‘I’ and inversely proportional to the distance ‘r’.
B = k * (I / r)
When multiple magnetic fields produced by different currents act simultaneously, their effects are superimposed. The resulting magnetic field can be represented as the vector sum of the individual fields:
B = B1 + B2 + … + Bn
Circulation of Magnetic Field
The circulation of a vector field along a closed line is a property of vector fields. To calculate the circulation of the magnetic field along an imaginary closed curve, divide the curve into elementary segments of length dS, multiply the length of the segment by the tangential component of the magnetic field, and then sum these products.
Circulation = Σ (B ⋅ dS ⋅ cos α) = Σ (B ⋅ dS)
If the tangential component of the field remains constant, the circulation is calculated as:
Circulation = (Tangential component) * (Length of curve)
Consider a trajectory consisting of two circular sections (A and C) concentric with the wire and two radial sections (B and D). Segments B and D do not contribute to the circulation because the magnetic field is perpendicular to the radial direction, making its tangential component zero. The contribution of semicircle C to the circulation is equal to that of semicircle A. Therefore, the circulation along the path ABCD is equal to the circulation along the circle of radius r.
In general, any curvilinear trajectory can be considered as consisting of very small circular and radial sections, such that the curve approximates the original path. The circulation along this line is equal to the circulation along the circle centered on the wire.
Ampere’s Law
CB = μ0 * I
Ampere’s Law states that the circulation of the magnetic field along a closed curve is proportional to the net current through the surface bounded by the curve.
When multiple wires pass through the surface, the circulation along the curve is proportional to the algebraic sum of the currents through the surface. Regardless of whether the wires are straight or curved, parallel or not, or whether the surface is flat or not, Ampere’s Law is valid in general. With some modification by Maxwell, it is one of the fundamental laws of electromagnetism.
According to Ampere’s Law, the circulation along a line is zero when the net current flowing through the area bounded by the line is zero. For example, the circulation along line ABCD in the picture below is zero. Segments B and D do not contribute to the circulation because the field is perpendicular to the radial direction. The circulation along semicircle C is negative because the tangential component of the magnetic field is opposite to the direction of travel, while the circulation along semicircle A is positive because the tangential component has the same direction as the displacement in that stretch.
Both circulations are offset by equal and opposite signs. So, even though this line is in a magnetic field, the circulation is zero when no current passes through. This reasoning can be extended to any curve by breaking it down into suitable radial and circular sections.