Question

Given \(\mathbf{H}=300 \mathbf{a}_{z} \cos \left(3 \times 10^{8} t-y\right) \mathrm{A} / \mathrm{m}\) in free space, find the emf developed in the general \(\mathbf{a}_{\phi}\) direction about the closed path having corners at \((a)(0,0,0),(1,0,0),(1,1,0)\), and \((0,1,0) ;(b)(0,0,0)(2 \pi, 0,0)\), \((2 \pi, 2 \pi, 0)\), and \((0,2 \pi, 0)\)

Short answer

Using Faraday's Law of electromagnetic induction, calculate the emf developed in the general a_φ direction about the closed paths: a) Path with corners at (0,0,0), (1,0,0), (1,1,0), and (0,1,0). b) Path with corners at (0,0,0), (2π, 0,0), (2π, 2π, 0), and (0,2π, 0). Recall that Faraday's Law is given by: ε = -dΦ/dt

Step-by-step solution

(Step 1: Write down the Faraday's Law)

Faraday's Law of electromagnetic induction states that the emf (electromotive force) induced in a closed loop is equal to the negative rate of change of magnetic flux with time. Mathematically, it is represented as: \(\epsilon = -\frac{d\Phi}{dt}\) where \(\epsilon\) is the emf and \(\Phi\) is the magnetic flux. We start by finding the magnetic flux through the closed loop in both cases (a) and (b). #Step1a# Calculate the magnetic flux for the first closed path Consider the closed path with corners at \((0,0,0),(1,0,0),(1,1,0)\), and \((0,1,0)\). We can represent the area vector of the loop (A) as: \(\mathbf{A} = (1-0)(1-0)\mathbf{a}_{z} = \mathbf{a}_{z}\) The magnetic flux Φ through this closed loop is given by the dot product of the magnetic field vector \(\mathbf{H}\) and the area vector \(\mathbf{A}\): \(\Phi = \int_{S} \mathbf{H} \cdot \mathbf{A} \, dS\) \(\Phi = \int_{S} \left[ 300 \mathbf{a}_{z} \cos \left(3 \times 10^{8} t-y\right) \right] \cdot \mathbf{a}_{z} \, dS\) #Step1b# Calculate the magnetic flux for the second closed path Now consider the closed path with corners at \((0,0,0),(2\pi, 0,0)\), \((2\pi, 2\pi, 0)\), and \((0,2\pi, 0)\). The area vector of this loop (A) can be represented as: \(\mathbf{A} = (2\pi - 0)(2\pi - 0)\mathbf{a}_{z} = 4\pi^2\mathbf{a}_{z}\) The magnetic flux Φ through this closed loop is given by the dot product of the magnetic field vector \(\mathbf{H}\) and the area vector \(\mathbf{A}\): \(\Phi = \int_{S} \mathbf{H} \cdot \mathbf{A} \, dS\) \(\Phi = \int_{S} \left[ 300 \mathbf{a}_{z} \cos \left(3 \times 10^{8} t-y\right) \right] \cdot 4\pi^2\mathbf{a}_{z} \, dS\) #Step2# Differentiate magnetic flux with respect to time Now we need to find the rate of change of magnetic flux with time in both scenarios. For the first scenario (a): \(\frac{d \Phi}{dt}=\int_{S} \frac{\partial}{\partial t}\left[ 300 \cos \left(3 \times 10^{8} t-y\right) \right] \, dS\) For the second scenario (b): \(\frac{d \Phi}{dt}=\int_{S} \frac{\partial}{\partial t}\left[ 300 \cos \left(3 \times 10^{8} t-y\right) \times 4\pi^2 \right] \, dS\) #Step3# Calculate the emf for both cases Finally, calculate the emf for both cases `a` and `b` using the formula \(\epsilon = -\frac{d\Phi}{dt}\): For the first scenario (a): \(\epsilon_a = -\frac{d\Phi_a}{dt}\) For the second scenario (b): \(\epsilon_b = -\frac{d\Phi_b}{dt}\)

Key concepts

Magnetic Flux

Magnetic flux is a measure of the number of magnetic field lines passing through a given area. Imagine a flat surface, like a piece of paper, and a magnetic field penetrating this surface. The amount of field lines crossing the paper defines the magnetic flux. It's often compared to the flow of water: the more water flowing through an area, the stronger the current, similarly, the more magnetic field lines, the greater the magnetic flux.

In mathematical terms, it is denoted by the Greek letter \(\Phi\) and is calculated by taking the dot product of the magnetic field \(\mathbf{H}\) with the area vector \(\mathbf{A}\). Mathematically, \(\Phi = \mathbf{H} \cdot \mathbf{A}\), where \(\mathbf{A}\) is perpendicular to the loop's surface. In an examination of a closed loop, the area through which the flux is calculated must be well-defined. This concept is vital for understanding how varying magnetic fields can induce an electromotive force (emf) within a closed circuit.

Electromotive Force (EMF)

Electromotive force, or emf, is not a force in the traditional sense. Instead, it refers to the potential difference that can produce an electric current in a circuit. It's like the 'pressure' that pushes the electric charges to move through the loop. An emf can be generated by a battery, a change in magnetic flux, or even a loop moving through a magnetic field.

The unit of emf is the volt, just like for potential difference. When Faraday's Law comes into play, it tells us that an emf is induced in a closed circuit when the magnetic flux through the area enclosed by the circuit changes. The idea is that the emf will cause electric charges to move, creating a current that's trying to counteract the change in magnetic flux — this is an expression of Lenz's Law, which forms the basis for many electrical generators and transformers.

Rate of Change of Magnetic Flux

The rate of change of magnetic flux is critical to Faraday's Law of electromagnetic induction. It's similar to how the acceleration of a car is crucial to understanding its motion: just as you'd look at how quickly a car speeds up to determine its behavior, scientists look at how quickly magnetic flux changes to predict the emf that will be generated.

The mathematical expression from Faraday's Law is \(\epsilon = -\frac{d\Phi}{dt}\). This equation tells us that the emf (\(\epsilon\)) induced in a circuit is equal to the negative rate of change of magnetic flux (\(-\frac{d\Phi}{dt}\)) through the circuit. The negative sign represents Lenz's Law, indicating that the induced emf creates a current whose own magnetic field opposes the original change in flux. Understanding this concept is fundamental when solving problems related to electromagnetic induction, as it allows one to calculate how a changing magnetic field can create an electric current.

Closed Loop Electromagnetic Induction

Closed loop electromagnetic induction is the phenomenon where an emf is induced in a closed conducting loop when the magnetic flux through the loop changes. The 'closed loop' aspect is essential: a complete path must exist for induced charges to circulate and create a current. This concept becomes particularly fascinating when the loop itself is stationary, but the magnetic field changes in strength or direction, or even when the loop moves within a steady magnetic field, altering the flux due to its change in position or orientation.

In a practical context, consider a coil of wire — a loop or several loops together — in a magnetic field. If the magnetic field changes, or if the coil moves, the magnetic flux through the coil changes, and according to Faraday's Law, an emf is induced. This simple principle underlies the operation of electric generators: they convert mechanical energy into electrical energy by rotating a coil within a magnetic field, thus changing the flux and inducing an emf.