Faraday’s Law of Electromagnetic Induction stands as one of the most fundamental principles in electrical engineering and physics. Discovered by Michael Faraday in 1831, this law describes how changing magnetic fields can generate electrical currents, enabling the operation of countless electrical devices from power generators to electric motors.
This blog explains the theoretical foundations, mathematical formulations and practical application to help electrical engineering students and professionals better understand this essential principle.
Faraday’s Law of Electromagnetic Induction
Faraday’s Law states that the electromotive force (EMF) induced in a conductor is directly proportional to the rate of change of magnetic flux through that conductor. The law emerged from a series of experiments conducted by Michael Faraday, who discovered that whenever the magnetic field around a conductor changes, an electric current is induced in that conductor. This phenomenon, known as electromagnetic induction.
The principle can be understood through a simple thought experiment: when a bar magnet moves toward or away from a coil of wire, the changing magnetic field induces an electric current in the coil. The faster the magnet moves, the greater the induced EMF becomes. Similarly, rotating a coil within a stationary magnetic field also produces a changing magnetic flux, thereby inducing an EMF.
Faraday’s First Law of Electromagnetic Induction
Faraday’s First Law provides a qualitative description of electromagnetic induction. It states that whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in the conductor. If the conductor forms a closed circuit, this induced EMF causes a current to flow, which is termed the induced current. The law emphasizes that the key requirement for induction is a change in the magnetic environment—whether through movement of the conductor, variation in the magnetic field strength, or alteration of the coil’s orientation.
Several methods can create the changing magnetic field necessary for induction:
- Moving a magnet toward or away from a coil
- Rotating the coil relative to a stationary magnetic field
- Changing the area of the coil placed in the magnetic field
- Varying the current in a nearby coil (mutual induction)
- Moving a conductor through a magnetic field
Faraday’s Second Law of Electromagnetic Induction
Faraday’s Second Law quantifies the relationship between induced EMF and magnetic flux change. It states that the magnitude of the induced EMF in a coil is equal to the rate of change of magnetic flux linkage through the coil. The flux linkage represents the product of the number of turns in the coil and the magnetic flux associated with each turn. This law provides the mathematical foundation for calculating induced voltages in practical applications.
The mathematical expression of Faraday’s Second Law is given by:
\( \varepsilon = -N \frac{d\Phi}{dt} \)
Where:
- \(\varepsilon\) is the induced electromotive force (EMF) in volts \((V)\)
- \(N\) is the number of turns in the coil
- \(\Phi\) is the magnetic flux in webers \((Wb)\)
- \(t\) is time in seconds (s)
- The negative sign represents Lenz’s Law
This equation shows several important relationships. First, increasing the number of turns in the coil proportionally increases the induced EMF. Second, increasing the magnetic field strength produces a greater induced voltage. Third, increasing the speed of relative motion between the coil and magnet results in a higher rate of flux change and consequently a larger induced EMF.
Magnetic Flux
Before fully appreciating Faraday’s Law, one must understand magnetic flux, denoted by \(\Phi_B\). Magnetic flux represents the total number of magnetic field lines passing through a given surface area. It quantifies the strength of the magnetic field penetrating a surface and serves as the fundamental quantity whose change induces EMF according to Faraday’s Law.
The magnetic flux through a surface is mathematically defined as:
\( \Phi_B = B \cdot A \cdot \cos(\theta) \)
Where:
- \(\Phi_B\) is the magnetic flux measured in webers \((Wb)\) or tesla-meter squared \((Tm^2)\)
- \(B\) is the magnetic field strength in teslas \((T)\)
- \(A\) is the area of the surface in square meters \((m^2)\)
- \(\theta\) is the angle between the magnetic field vector and the normal (perpendicular) to the surface
This equation demonstrates that magnetic flux depends not only on the field strength and area but also critically on the orientation of the surface relative to the field. When the magnetic field is perpendicular to the surface \((\theta=0^o)\), the flux reaches its maximum value of \(B\times A\) because \((cos(0^o)=1\). Conversely, when the field is parallel to the surface \((\theta=90^o\), the flux equals zero since \((cos(90^o)=0\).
For non-uniform magnetic fields or irregularly shaped surfaces, the total flux must be calculated using integration:
\( \Phi_B = \int_S \vec{B} \cdot d\vec{A} \)
This integral sums the contributions of the magnetic field over all differential area elements of the surface.
The Maxwell-Faraday Equation
In its most fundamental form, Faraday’s Law appears as one of Maxwell’s four equations describing electromagnetism. The Maxwell-Faraday equation expresses the relationship between time-varying magnetic fields and induced electric fields in differential form:
\( \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \)
This equation states that a time-varying magnetic field always accompanies a spatially varying, circulating electric field. The curl of the electric field equals the negative time derivative of the magnetic field. This formulation applies to the fields themselves and does not require the presence of a physical circuit.
The integral form, obtained through the Kelvin-Stokes theorem, relates the circulation of the electric field around a closed loop to the rate of change of magnetic flux through the enclosed area:
\( \oint_{\partial \Sigma} \vec{E} \cdot d\vec{l} = -\iint_{\Sigma} \frac{\partial \vec{B}}{\partial t} \cdot d\vec{A} \)
This integral form directly connects to the practical measurement of induced EMF around a conducting loop.
Lenz’s Law and the Negative Sign
The negative sign in Faraday’s Law equation carries profound physical significance, from Lenz’s Law. Formulated by the German physicist Heinrich Lenz in 1834, this law specifies the direction of the induced current. Lenz’s Law states that the induced current flows in such a direction that its own magnetic field opposes the change in magnetic flux that produced it.
If the induced current’s magnetic field aided rather than opposed the flux change, energy would be created spontaneously, violating the law of conservation of energy. Lenz’s Law ensures that work must be done to change the magnetic flux through a circuit.
Practically, Lenz’s Law provides a method to determine current direction. When a north pole of a magnet approaches a coil, the induced current creates a magnetic field making the near side of the coil also a north pole, thereby repelling the approaching magnet. When the magnet is withdrawn, the induced current reverses, making the near side a south pole to attract the receding magnet.
Applications of Faraday’s Law in Electrical Engineering
Faraday’s Law underlies the operation of virtually all electromagnetic devices used in modern electrical engineering. From massive power station generators to miniature sensors, this principle enables the conversion between mechanical and electrical energy, voltage transformation, and countless other functions essential to contemporary technology.
1. Electric Generators
Electric generators represent perhaps the most important application of Faraday’s Law, converting mechanical energy into electrical energy to power modern civilization. A generator operates by rotating a coil of wire within a stationary magnetic field or rotating magnets around stationary coils. As the coil rotates, the magnetic flux through it continuously changes, inducing an alternating EMF according to Faraday’s Law.
2. Transformers
Transformers rely fundamentally on Faraday’s Law and the principle of mutual induction to transfer electrical energy between circuits while changing voltage levels. A transformer consists of two coils—a primary coil and a secondary coil—wound around a common ferromagnetic core, typically made of laminated steel.
When alternating current flows through the primary coil, it creates a time-varying magnetic field that the core channels toward the secondary coil. This changing magnetic flux induces an EMF in the secondary coil according to Faraday’s Law. The voltage ratio between the coils depends directly on their turn ratio:
\( \frac{V_2}{V_1} = \frac{N_2}{N_1} \)
Where \(V_1\) and \(V_2\) are the primary and secondary voltages, and 
and are the respective number of turns. A transformer with more turns in the secondary than the primary acts as a step-up transformer, increasing voltage. Conversely, fewer secondary turns create a step-down transformer, decreasing voltage.
3. Electric Motors and Back EMF
Electric motors convert electrical energy into mechanical energy, functioning essentially as generators in reverse. While motors primarily operate through the magnetic force on current-carrying conductors, Faraday’s Law plays a crucial role through the phenomenon of back EMF.
As a motor’s coil rotates in the magnetic field, the changing flux through the coil induces an EMF according to Faraday’s Law. This induced EMF opposes the applied voltage that drives the motor, hence the term “back EMF”.
The back EMF is given by:
\( E_b = K_e \cdot \omega \)
Where \(E_b\) is the back EMF, \(K_e\) is the back EMF constant (specific to the motor design), and \(\omega\) is the angular velocity of the rotor. As the motor speed increases, the back EMF increases proportionally.
4. Induction Cooking and Heating
Induction cooktops represent a modern consumer application of Faraday’s Law, offering efficient and precise cooking. An induction cooktop contains a coil through which high-frequency alternating current flows, creating a rapidly changing magnetic field. When a ferromagnetic cooking vessel (such as cast iron or magnetic stainless steel) is placed on the cooktop, the changing magnetic field induces eddy currents in the metal base of the pan according to Faraday’s Law.
5. Wireless Charging and Power Transfer
Wireless charging technology for smartphones, electric vehicles, and other devices operates on the principle of electromagnetic induction described by Faraday’s Law. A charging station contains a transmitter coil carrying alternating current, which generates a time-varying magnetic field. When a compatible device with a receiver coil is placed nearby, the changing magnetic flux induces an EMF in the receiver coil. This induced voltage charges the device’s battery without any physical electrical connection.
6. Sensors and Measurement Devices
Numerous sensor technologies exploit electromagnetic induction. Metal detectors, commonly seen at airports and archaeological sites, use Faraday’s Law to detect metallic objects. A transmitter coil in the detector creates a time-varying magnetic field. When metal is nearby, the changing field induces eddy currents in the metal, which in turn produce their own magnetic field.
Magnetic stripe cards, including credit cards and access cards, store data in magnetic patterns. When the card is swiped through a reader, the changing magnetic field as the stripe moves past the reading head induces voltage variations in the reader’s coil according to Faraday’s Law. These voltage patterns encode the stored information.
Electromagnetic flowmeters measure the flow rate of conductive fluids using Faraday’s Law. A magnetic field is applied perpendicular to the fluid flow direction. As the conductive fluid moves through the field, an EMF is induced proportional to the flow velocity. Measuring this induced voltage allows calculation of the volumetric flow rate.
7. Medical Applications: MRI and Magnetic Therapy
Magnetic Resonance Imaging (MRI) machines utilize principles of electromagnetic induction for medical diagnostics. While the primary physics of MRI involves nuclear magnetic resonance, the detection of signals from hydrogen nuclei in the body relies on induced currents in receiver coils. The changing magnetic fields produced by precessing hydrogen nuclei induce tiny voltages in the receiver coils according to Faraday’s Law, which are then amplified and processed to construct detailed images of internal body structures.
Self-Inductance and Mutual Inductance
Beyond the basic formulation of Faraday’s Law, the concepts of self-inductance and mutual inductance provide deeper understanding of electromagnetic induction in coils and coupled circuits.
Self-Inductance
Self-inductance describes how a changing current in a coil induces an EMF in that same coil. When current through a coil changes, the magnetic field it produces also changes. This changing magnetic field induces an EMF in the coil itself that opposes the current change—a phenomenon described by Lenz’s Law.
The self-induced EMF is given by:
\( \varepsilon = -L \frac{dI}{dt} \)
Where \(L\) is the self-inductance (or simply inductance) of the coil, measured in henries \((H)\), and \(\frac{dI}{dt}\) is the rate of change of current.
Mutual Inductance
Mutual inductance describes the interaction between two nearby coils. When current in one coil (the primary) changes, its changing magnetic field induces an EMF in the nearby coil (the secondary). This is the principle underlying transformer operation.
The mutual inductance \(M\) is defined by:
\( \varepsilon_2 = -M \frac{dI_1}{dt} \)
Where \(\varepsilon_2\) is the induced EMF in the secondary coil due to the changing current \(\frac{dI_1}{dt}\) in the primary coil.
For two closely coupled coils, the mutual inductance relates to their self-inductances through:
\( M = K\sqrt{L_1 L_2} \)
Where \(L_1\) and \(L_2\) are the self-inductances of the two coils, and \(K\) is the coupling coefficient \((0 ≤ K ≤ 1)\). The coupling coefficient indicates how effectively the magnetic flux from one coil links with the other. Perfect coupling \((K = 1)\) means all flux from one coil passes through the other, while K = 0 indicates no coupling.
Practical Calculations and Examples
Example 1: Induced EMF in a Solenoid
Consider a solenoid with 200 turns per meter, a cross-sectional area of \(2.0\times 10^-3\,m^2\), and a magnetic field that changes from 0.30 T to 0.70 T in 0.50 seconds. Calculate the magnitude of the induced EMF.
Solution:
First, determine the total number of turns if we consider a solenoid of length 1 meter: \(N = 200\) turns.
Calculate the change in magnetic flux:
\( \Delta \Phi = \Delta B \times A \)
\(= (0.70 – 0.30) \times 2.0 \times 10^{-3}\)
\(= 0.40 \times 2.0 \times 10^{-3}\)
\( = 8.0 \times 10^{-4} Wb \)
Apply Faraday’s Law:
\( |\varepsilon| = N \frac{\Delta \Phi}{\Delta t} \)
\(= 200 \times \frac{8.0 \times 10^{-4}}{0.50}\)
\( = 200 \times 1.6 \times 10^{-3} = 0.32 V \)
The magnitude of the induced EMF is \(0.32\) volts.
Example 2: Magnetic Flux Through a Coil
A circular coil has a radius of 25 cm and is placed perpendicular to a uniform magnetic field of 20 T. Calculate the magnetic flux through the coil.
Solution:
The area of the circular coil is:
\( A = \pi r^2 = \pi \times (0.25)^2 = 0.196 m^2 \)
Since the magnetic field is perpendicular to the coil, \(\theta=0^o\) and \(cos(0^o)=1\).
The magnetic flux is:
\( \Phi_B = B \times A \times \cos(\theta) = 20 \times 0.196 \times 1 = 3.93 Wb \)
The magnetic flux through the coil is 3.93 webers.
Example 3: Transformer Voltage Calculation
A transformer has 500 turns in its primary coil and 2000 turns in its secondary coil. If the primary voltage is 120 V, what is the secondary voltage?
Solution:
Using the transformer equation:
\( \frac{V_2}{V_1} = \frac{N_2}{N_1} \)
Solving for \(V_2\):
\( V_2 = V_1 \times \frac{N_2}{N_1} = 120 \times \frac{2000}{500} = 120 \times 4 = 480 V \)
The secondary voltage is 480 V. This is a step-up transformer, increasing voltage by a factor of 4.
Example 4: EMF in a Moving Conductor
A straight conductor of length 0.5 m moves at a velocity of 10 m/s perpendicular to a uniform magnetic field of 0.8 T. Calculate the induced EMF.
Solution:
For a conductor moving perpendicular to a magnetic field, the induced EMF is:
\( \varepsilon = B \times L \times v = 0.8 \times 0.5 \times 10 = 4.0 V \)
The induced EMF in the moving conductor is 4.0 volts.
Induced EMF Calculator (Faraday’s Law)
This calculator computes the induced EMF using Faraday’s Law formula: \(\varepsilon=-N\times \frac{\Delta\Phi}{\Delta t}\)
Induced EMF Calculator
Calculate induced EMF using Faraday’s Law: ε = -N × (ΔΦ/Δt)
Magnetic Flux Calculator
This calculator computes magnetic flux using the formula: \(\Phi=B\times A\times cos(\theta)\)
Magnetic Flux Calculator
Calculate magnetic flux: Φ = B × A × cos(θ)
EMF in Moving Conductor Calculator
This calculator computes EMF in a conductor moving through a magnetic field: \(EMF=B\times L\times v\)
Moving Conductor EMF Calculator
Calculate EMF in moving conductor: EMF = B × L × v
Conclusion
Faraday’s Law of Electromagnetic Induction stands as one of the most profound and practically significant discoveries in the history of physics and electrical engineering. Michael Faraday’s experimental insights, formalized into mathematical relationships, have enabled virtually all modern electrical technology—from the massive generators that power cities to the tiny sensors in smartphones. The principle that a changing magnetic field induces an electromotive force provides the foundation for understanding transformers, motors, generators, induction heating, wireless charging, and countless other applications.