What is Maxwell's second law?
2 Answers
Maxwell's second law is also known as **Faraday's Law of Induction**. It states that the electromotive force (EMF) induced in a closed circuit is proportional to the negative rate of change of the magnetic flux through the circuit. Mathematically, it is expressed as:
\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]
where \(\mathcal{E}\) is the induced EMF, and \(\Phi_B\) is the magnetic flux through the circuit. This law is a fundamental principle behind electromagnetic induction, which is the basis for many electrical devices, such as transformers and generators.
\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]
where \(\mathcal{E}\) is the induced EMF, and \(\Phi_B\) is the magnetic flux through the circuit. This law is a fundamental principle behind electromagnetic induction, which is the basis for many electrical devices, such as transformers and generators.
Maxwell's Second Law, often referred to as one of Maxwell's equations, is more accurately called **Faraday's Law of Induction**. This law is fundamental in electromagnetism and describes how a changing magnetic field can induce an electric field. Here’s a detailed explanation:
### Faraday’s Law of Induction
**Mathematical Formulation:**
Faraday’s Law of Induction is expressed mathematically as:
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
Here:
- \( \nabla \times \mathbf{E} \) represents the curl of the electric field \( \mathbf{E} \).
- \( \frac{\partial \mathbf{B}}{\partial t} \) is the partial derivative of the magnetic field \( \mathbf{B} \) with respect to time \( t \).
**Physical Interpretation:**
1. **Induced Electric Field:** Faraday's Law states that a time-varying magnetic field induces a circulating electric field. The curl of the electric field is proportional to the negative rate of change of the magnetic field. This means that a changing magnetic field creates an electric field that circulates around the area where the magnetic field is changing.
2. **Electromagnetic Induction:** This principle is the basis for many electrical technologies, such as electric generators and transformers. For example, in a generator, mechanical energy is used to change the magnetic field, which then induces an electric current in a coil.
3. **Lenz's Law:** The negative sign in the equation indicates the direction of the induced electric field. According to Lenz's Law, the direction of the induced electric field (and the resulting induced current, if there's a closed loop) is such that it opposes the change in the magnetic field that produced it. This is a manifestation of the principle of conservation of energy.
**Integral Form:**
Faraday's Law can also be expressed in integral form:
\[ \oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_{S} \mathbf{B} \cdot d\mathbf{A} \]
Here:
- The left side, \( \oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} \), represents the electromotive force (EMF) around a closed loop \( \partial S \).
- The right side, \( -\frac{d}{dt} \int_{S} \mathbf{B} \cdot d\mathbf{A} \), represents the rate of change of the magnetic flux through the surface \( S \) bounded by the loop.
**Applications:**
1. **Electric Generators:** As the magnetic field around a coil changes (typically due to mechanical rotation), an electric current is induced in the coil due to Faraday's Law.
2. **Transformers:** Faraday’s Law explains how alternating current in one coil (primary) induces a voltage in another coil (secondary) through a changing magnetic field.
3. **Induction Cooktops:** A changing magnetic field generates eddy currents in the cookware, which in turn heats it up due to resistance.
Faraday’s Law of Induction is a cornerstone of electromagnetism, linking electric and magnetic fields and providing insight into how they interact dynamically.
### Faraday’s Law of Induction
**Mathematical Formulation:**
Faraday’s Law of Induction is expressed mathematically as:
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
Here:
- \( \nabla \times \mathbf{E} \) represents the curl of the electric field \( \mathbf{E} \).
- \( \frac{\partial \mathbf{B}}{\partial t} \) is the partial derivative of the magnetic field \( \mathbf{B} \) with respect to time \( t \).
**Physical Interpretation:**
1. **Induced Electric Field:** Faraday's Law states that a time-varying magnetic field induces a circulating electric field. The curl of the electric field is proportional to the negative rate of change of the magnetic field. This means that a changing magnetic field creates an electric field that circulates around the area where the magnetic field is changing.
2. **Electromagnetic Induction:** This principle is the basis for many electrical technologies, such as electric generators and transformers. For example, in a generator, mechanical energy is used to change the magnetic field, which then induces an electric current in a coil.
3. **Lenz's Law:** The negative sign in the equation indicates the direction of the induced electric field. According to Lenz's Law, the direction of the induced electric field (and the resulting induced current, if there's a closed loop) is such that it opposes the change in the magnetic field that produced it. This is a manifestation of the principle of conservation of energy.
**Integral Form:**
Faraday's Law can also be expressed in integral form:
\[ \oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_{S} \mathbf{B} \cdot d\mathbf{A} \]
Here:
- The left side, \( \oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} \), represents the electromotive force (EMF) around a closed loop \( \partial S \).
- The right side, \( -\frac{d}{dt} \int_{S} \mathbf{B} \cdot d\mathbf{A} \), represents the rate of change of the magnetic flux through the surface \( S \) bounded by the loop.
**Applications:**
1. **Electric Generators:** As the magnetic field around a coil changes (typically due to mechanical rotation), an electric current is induced in the coil due to Faraday's Law.
2. **Transformers:** Faraday’s Law explains how alternating current in one coil (primary) induces a voltage in another coil (secondary) through a changing magnetic field.
3. **Induction Cooktops:** A changing magnetic field generates eddy currents in the cookware, which in turn heats it up due to resistance.
Faraday’s Law of Induction is a cornerstone of electromagnetism, linking electric and magnetic fields and providing insight into how they interact dynamically.