Problem 2176 - JEE Physics

Question

What is the name associated with the equation \(E^{\longrightarrow} \cdot d t^{-}=-(d \Phi \beta / d t)\) (A) Gauss law for electricity (B) Gauss law for magnetism (C) ampere's law (D) faraday's law

Step-by-Step Solution

Answer

The given equation \(E^{\longrightarrow} \cdot dt^{-} = -(\frac{d\Phi \beta}{dt})\) is related to (D) Faraday's law, as it deals with the relationship between the electric field and the rate of change of magnetic flux.

Step 1: Analyze the given equation

The given equation is: \(E^{\longrightarrow} \cdot dt^{-} = -(\frac{d\Phi \beta}{dt})\) where \(E^{\longrightarrow}\) represents the electric field, \(dt^{-}\) is a change in time, and \(d\Phi \beta / dt\) represents the rate of change of magnetic flux. This equation indicates that the electric field is related to the rate of change of magnetic flux.

Step 2: Comparing with the given options

Now, let's compare the given equation with the laws mentioned in the options: (A) Gauss's law for electricity: It deals with the electric flux through a closed surface and the enclosed electric charge, not the relation between the electric field and magnetic flux. (B) Gauss's law for magnetism: It states that the net magnetic flux through a closed surface is zero. This law is also not related to the given equation. (C) Ampere's law: It relates the circulation of the magnetic field around a closed loop to the net current passing through the loop. It also doesn't match with our equation. (D) Faraday's law: Faraday's law of electromagnetic induction states that the electromotive force (EMF) induced in a circuit is proportional to the rate of change of magnetic flux. It can be mathematically represented as: \(EMF = -\frac{d\Phi \beta}{dt}\) Comparing the given equation with Faraday's law, we can see a clear connection between them since both deal with the relationship between the electric field and the rate of change of magnetic flux.

Step 3: Conclusion

Based on the comparison made in step 2, it is evident that the given equation is related to (D) Faraday's law.

Key Concepts

Electromagnetic Induction

Electromagnetic induction is a fundamental principle that explains how a changing magnetic field can generate an electric field. This concept was discovered by Michael Faraday in the 19th century. Let's break it down simply:

• When a magnetic field around a conductor (like a wire) changes, it induces a voltage or electromotive force (EMF) across the conductor.
• This induced voltage can create an electric current if there is a closed loop path.
• The faster the magnetic field changes, the greater the induced EMF will be.

Electromagnetic induction is the working principle behind many electrical devices like transformers, electric generators, and induction cooktops. It’s amazing how this simple concept is used in so many applications!
Keep in mind, electromagnetic induction is all about the interaction between electricity and magnetism. Understanding this principle allows us to see how electrical currents can be generated and manipulated.

Electric Field

The electric field is a crucial concept in physics that helps describe how electric charges interact. Think of it as an invisible force field surrounding electric charges. This field influences any other charges that enter its area of influence. Here are some essential points:

• It is represented by vectors, showing both the direction and magnitude of the force experienced by a positive test charge.
• The strength of the electric field is measured in volts per meter (V/m).
• Mathematically, it is described by the equation \( E = F / q \), where \( E \) is the electric field, \( F \) is the force, and \( q \) is the charge.

Electric fields are fundamental in electromagnetism, influencing how electric devices and systems function. You can find electric fields in things like capacitors, which store energy, or in our atmosphere, where lightning happens as a natural illustration of electric field discharge. Remember, the stronger the field, the more significant the effect on other charges it will have.

Magnetic Flux

Magnetic flux measures the total magnetic field that passes through a given area. It's a vital concept in understanding how magnetic fields interact with materials and space. Some key aspects include:

• It is represented by the symbol \( \Phi \) (Phi) and measured in Weber (Wb).
• Magnetic flux depends on two main factors: the strength of the magnetic field and the area it penetrates.
• The angle at which the field lines pass through the area also affects the flux.

In equations, magnetic flux can be calculated using \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field strength, \( A \) is the area in question, and \( \theta \) is the angle relative to the perpendicular to the surface.Magnetic flux is crucial for devices like inductors and transformers, and it’s directly related to how electricity is generated via electromagnetic induction. By understanding magnetic flux, we can better grasp how changes in magnetic environments result in electric phenomena.

Rate of Change of Magnetic Flux

The rate of change of magnetic flux is essential in understanding Faraday's Law of electromagnetic induction. It's all about how fast the magnetic flux changes over time, which results in inducing an electromotive force (EMF).

• The concept is expressed mathematically as \( \frac{d\Phi}{dt} \), representing how magnetic flux \( \Phi \) varies with time \( t \).
• This rate of change is the key factor in determining the magnitude of the induced EMF - faster changes mean stronger EMF.
• Considering the sign, a negative rate of change means the induced EMF opposes the change through Lenz's law.

In practical applications, the rate of change of magnetic flux is pivotal for designing electric generators, where rotating coils in magnetic fields produce electricity. This principle also underlies many modern technologies, showing how essential understanding changing magnetic environments is to harnessing electromagnetic energy. Always remember, it's not just the presence of flux that matters, but how it changes that does!