The Induced Voltage Formula, expressed as \ ( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \), calculates the electromotive force (EMF) generated in a conductor due to a changing magnetic flux. This formula is a direct application of Faraday’s Law of Electromagnetic Induction and forms a core topic in NCERT Class 12 Physics, Chapter 6. It is equally critical for JEE Main, JEE Advanced, and NEET aspirants, as electromagnetic induction questions appear every year in these exams. This article covers the formula derivation, variables, a complete formula sheet, three solved examples, CBSE exam tips, common mistakes, and JEE/NEET applications.
Key Induced Voltage Formulas at a Glance
Quick reference for the most important induced voltage and electromagnetic induction formulas.
- Faraday’s Law (induced EMF): \( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \)
- Magnetic flux: \( \Phi = B A \cos\theta \)
- Motional EMF: \( \varepsilon = Blv \)
- Induced EMF in rotating coil: \( \varepsilon = N B A \omega \sin(\omega t) \)
- Self-induced EMF: \( \varepsilon = -L \frac{dI}{dt} \)
- Mutually induced EMF: \( \varepsilon = -M \frac{dI}{dt} \)
- Induced current: \( I = \frac{\varepsilon}{R} \)
What is the Induced Voltage Formula?
The Induced Voltage Formula defines the electromotive force (EMF) produced in a closed conductor or coil when the magnetic flux through it changes with time. This phenomenon is called electromagnetic induction. It was discovered by Michael Faraday in 1831 and is described mathematically by Faraday’s Law.
According to Faraday’s Law, the magnitude of the induced EMF in a coil is directly proportional to the rate of change of magnetic flux through the coil. The negative sign in the formula arises from Lenz’s Law, which states that the induced EMF always opposes the change that causes it.
This concept is covered in NCERT Class 12 Physics, Chapter 6 — Electromagnetic Induction. It is one of the most important chapters for CBSE Board exams and competitive examinations. The induced voltage can be generated by changing the magnetic field strength, changing the area of the coil, or changing the angle between the coil and the field.
The Induced Voltage Formula applies to transformers, electric generators, induction motors, and many modern electronic devices. Understanding this formula is essential for any student aiming to excel in physics at the Class 12 level and beyond.
Induced Voltage Formula — Expression and Variables
The primary form of the Induced Voltage Formula, derived from Faraday’s Law, is:
\[ \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \]
Here, the negative sign indicates that the induced EMF opposes the change in flux (Lenz’s Law). For a single loop, \( N = 1 \), giving \( \varepsilon = -\frac{\Delta \Phi}{\Delta t} \).
The magnetic flux itself is defined as:
\[ \Phi = B A \cos\theta \]
Substituting, the induced voltage can also be written as:
\[ \varepsilon = -N \frac{\Delta (B A \cos\theta)}{\Delta t} \]
| Symbol | Quantity | SI Unit |
|---|---|---|
| \( \varepsilon \) | Induced EMF (Induced Voltage) | Volt (V) |
| \( N \) | Number of turns in the coil | Dimensionless |
| \( \Delta \Phi \) | Change in magnetic flux | Weber (Wb) |
| \( \Delta t \) | Change in time | Second (s) |
| \( B \) | Magnetic field strength | Tesla (T) |
| \( A \) | Area of the coil | Square metre (m²) |
| \( \theta \) | Angle between field and normal to coil | Degree or Radian |
Derivation of the Induced Voltage Formula
Faraday observed that whenever the magnetic flux through a coil changes, a voltage is induced in the coil. He defined magnetic flux as \( \Phi = B A \cos\theta \).
Step 1: Consider a coil with \( N \) turns placed in a magnetic field \( B \).
Step 2: The total flux linkage through the coil is \( N\Phi \).
Step 3: Faraday’s Law states that the induced EMF equals the negative rate of change of total flux linkage:
\[ \varepsilon = -\frac{d(N\Phi)}{dt} = -N\frac{d\Phi}{dt} \]
Step 4: For a finite time interval \( \Delta t \), the formula becomes \( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \).
The negative sign is the mathematical expression of Lenz’s Law. It ensures energy conservation in electromagnetic systems.
Complete Electromagnetic Induction Formula Sheet
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Faraday’s Law (Induced Voltage) | \( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \) | ε = EMF, N = turns, ΔΦ = flux change, Δt = time | Volt (V) | Class 12, Ch 6 |
| Magnetic Flux | \( \Phi = B A \cos\theta \) | B = field, A = area, θ = angle | Weber (Wb) | Class 12, Ch 6 |
| Motional EMF | \( \varepsilon = Blv \) | B = field, l = length, v = velocity | Volt (V) | Class 12, Ch 6 |
| EMF in Rotating Coil | \( \varepsilon = N B A \omega \sin(\omega t) \) | N = turns, B = field, A = area, ω = angular frequency | Volt (V) | Class 12, Ch 7 |
| Peak EMF | \( \varepsilon_0 = N B A \omega \) | N = turns, B = field, A = area, ω = angular frequency | Volt (V) | Class 12, Ch 7 |
| Self-Induced EMF | \( \varepsilon = -L \frac{dI}{dt} \) | L = self-inductance, I = current, t = time | Volt (V) | Class 12, Ch 6 |
| Mutually Induced EMF | \( \varepsilon = -M \frac{dI}{dt} \) | M = mutual inductance, I = current, t = time | Volt (V) | Class 12, Ch 6 |
| Induced Current | \( I = \frac{\varepsilon}{R} \) | ε = induced EMF, R = resistance | Ampere (A) | Class 12, Ch 6 |
| Self-Inductance of Solenoid | \( L = \mu_0 n^2 V \) | μ&sub0; = permeability, n = turns per unit length, V = volume | Henry (H) | Class 12, Ch 6 |
| Energy Stored in Inductor | \( U = \frac{1}{2} L I^2 \) | L = inductance, I = current | Joule (J) | Class 12, Ch 6 |
Induced Voltage Formula — Solved Examples
Example 1 (Class 10-11 Level)
Problem: A coil of 50 turns is placed in a magnetic field. The magnetic flux through the coil changes from 0.02 Wb to 0.06 Wb in 0.5 seconds. Calculate the induced EMF.
Given:
- Number of turns, \( N = 50 \)
- Initial flux, \( \Phi_1 = 0.02 \) Wb
- Final flux, \( \Phi_2 = 0.06 \) Wb
- Time interval, \( \Delta t = 0.5 \) s
Step 1: Write the Induced Voltage Formula: \( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \)
Step 2: Calculate the change in flux: \( \Delta \Phi = \Phi_2 – \Phi_1 = 0.06 – 0.02 = 0.04 \) Wb
Step 3: Substitute values: \( \varepsilon = -50 \times \frac{0.04}{0.5} \)
Step 4: Calculate: \( \varepsilon = -50 \times 0.08 = -4 \) V
The magnitude of the induced EMF is 4 V. The negative sign indicates the direction opposes the flux increase (Lenz’s Law).
Answer
Induced EMF = 4 V (in magnitude)
Example 2 (Class 12 / CBSE Board Level)
Problem: A straight conductor of length 0.5 m moves with a velocity of 4 m/s perpendicular to a uniform magnetic field of 0.8 T. Calculate the motional EMF induced in the conductor.
Given:
- Length of conductor, \( l = 0.5 \) m
- Velocity, \( v = 4 \) m/s
- Magnetic field, \( B = 0.8 \) T
- Angle between velocity and field = 90° (perpendicular)
Step 1: Use the motional EMF formula: \( \varepsilon = Blv \)
Step 2: Substitute values: \( \varepsilon = 0.8 \times 0.5 \times 4 \)
Step 3: Calculate: \( \varepsilon = 0.8 \times 2 = 1.6 \) V
Step 4: Since the conductor moves perpendicular to the field, no angular correction is needed. The induced voltage is 1.6 V.
Answer
Motional EMF = 1.6 V
Example 3 (JEE/NEET Level)
Problem: A rectangular coil of 200 turns, each of area 0.1 m², rotates at an angular frequency of 50 rad/s in a uniform magnetic field of 0.5 T. The resistance of the coil is 100 Ω. Find (a) the peak induced EMF and (b) the peak induced current.
Given:
- Number of turns, \( N = 200 \)
- Area, \( A = 0.1 \) m²
- Angular frequency, \( \omega = 50 \) rad/s
- Magnetic field, \( B = 0.5 \) T
- Resistance, \( R = 100 \) Ω
Step 1: Use the peak EMF formula: \( \varepsilon_0 = N B A \omega \)
Step 2: Substitute values: \( \varepsilon_0 = 200 \times 0.5 \times 0.1 \times 50 \)
Step 3: Calculate peak EMF: \( \varepsilon_0 = 200 \times 0.5 \times 5 = 200 \times 2.5 = 500 \) V
Step 4: Use the induced current formula: \( I_0 = \frac{\varepsilon_0}{R} = \frac{500}{100} = 5 \) A
Step 5: The instantaneous EMF at time \( t \) is \( \varepsilon = 500 \sin(50t) \) V.
Answer
(a) Peak induced EMF = 500 V (b) Peak induced current = 5 A
CBSE Exam Tips 2025-26
- Always state Lenz’s Law: In CBSE 2025-26 board exams, questions on induced EMF often carry marks for mentioning the negative sign and explaining Lenz’s Law. Never skip this step.
- Distinguish between EMF and voltage: Induced EMF is an open-circuit quantity. Induced voltage across a load differs when current flows. We recommend practising both scenarios.
- Memorise the motional EMF formula: \( \varepsilon = Blv \) is frequently asked as a 2-mark short answer. Know when to use it versus Faraday’s Law.
- Unit consistency: Always convert area to m², field to Tesla, and time to seconds before substituting. Unit errors are a common source of lost marks.
- Rotating coil questions: For AC generator problems, remember that peak EMF is \( \varepsilon_0 = NBA\omega \). CBSE frequently asks students to derive this expression.
- Draw diagrams: In 3-mark and 5-mark questions, a labelled diagram of the coil in the magnetic field earns 1 mark. Our experts suggest always including it.
Common Mistakes to Avoid
- Ignoring the negative sign: Many students drop the negative sign in \( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \). This sign carries physical meaning (Lenz’s Law) and is required in derivation-based questions.
- Confusing flux with field: Magnetic flux \( \Phi = BA\cos\theta \) is not the same as the magnetic field \( B \). Always compute flux first before applying Faraday’s Law.
- Wrong angle in flux calculation: The angle \( \theta \) in \( \Phi = BA\cos\theta \) is the angle between the magnetic field vector and the normal to the coil’s surface. Students often use the angle with the plane of the coil instead, which gives \( \sin\theta \) instead of \( \cos\theta \).
- Forgetting the number of turns: For a coil with \( N \) turns, the total flux linkage is \( N\Phi \), not just \( \Phi \). Omitting \( N \) is a very common error in numerical problems.
- Using wrong formula for motional EMF: The formula \( \varepsilon = Blv \) applies only when the conductor moves perpendicular to both its own length and the magnetic field. For other angles, the formula becomes \( \varepsilon = Blv\sin\theta \).
JEE/NEET Application of Induced Voltage Formula
In our experience, JEE aspirants encounter the Induced Voltage Formula in multiple forms across different problem types. Understanding these patterns is key to scoring full marks.
Pattern 1: Changing Area (Sliding Rod Problems)
JEE Main frequently features problems where a conducting rod slides along parallel rails in a magnetic field. The area of the circuit changes with time. The induced EMF is calculated as:
\[ \varepsilon = B l v \]
Here, \( l \) is the separation between the rails and \( v \) is the velocity of the rod. These problems often combine with force and power calculations. The braking force on the rod is \( F = BIl = \frac{B^2 l^2 v}{R} \).
Pattern 2: Changing Magnetic Field
NEET and JEE Advanced problems often involve a coil in a time-varying magnetic field, such as \( B = B_0 \sin(\omega t) \). Students must differentiate the flux with respect to time to find the induced EMF. This tests calculus skills directly.
\[ \varepsilon = -N A \frac{dB}{dt} = -N A B_0 \omega \cos(\omega t) \]
Pattern 3: Rotating Coil (AC Generator)
Both JEE and NEET include AC generator problems. The coil rotates at angular velocity \( \omega \), and the flux changes sinusoidally. The peak EMF is \( \varepsilon_0 = NBA\omega \). Students must identify peak, RMS, and instantaneous values. In our experience, JEE aspirants who practise all three forms of the rotating coil formula consistently score better in the Alternating Current chapter as well.
We recommend solving at least 20 previous year JEE and NEET questions on electromagnetic induction. Focus on problems that combine the Induced Voltage Formula with Lenz’s Law, energy methods, and circuit analysis.
FAQs on Induced Voltage Formula
For more related physics formulas, explore our Complete Physics Formula Hub. You may also find these articles helpful: Electric Flux Formula, Angular Speed Formula, and Average Acceleration Formula. For the official NCERT syllabus, refer to the NCERT official website.