Electricity and Magnetism. (Mag Field)

[Partap03]

Okay, I will be honest, this is a homework assignment. For that matter this is the last homework assignment which is very crucial to preparing for the final. But the problem is I have 2 more finals for which I need to study for. So if you can please help me out and give a detailed solution I will appreciate it. Thanks in advance.

A circular wire loop (radius a = 1m) has a resistance of 5.0 ohms and lies in the x-y plane. A spatially uniform but time-dependent magnetic field exists throughout the region. What is the magnetic flux passing through the loop at t = 2.0s? Show the direction of the current induced in the loop at t = 2.0s. B = 1.0 T sin(wt) [k hat]; w = 1.0 rad/s.

 

[jbsteele]

The magnetic flux through the loop at t = 2.0s is given by Φ = BAcosθ = πa2Bcosθ, where θ is the angle between the normal to the plane of the loop and the magnetic field. At t = 2.0s, the magnetic field is B = 1.0 T sin (wt) [k hat], so the magnetic flux through the loop is Φ = πa2Bcosθ = πa2(1.0 T sin (wt))cosθ. The direction of the current induced in the loop at t = 2.0s can be determined using Faraday's law of induction, which states that the rate of change of the magnetic flux through a loop is equal to the emf induced in the loop. Since the magnetic field is changing in time, an emf will be induced in the loop and a current will flow. The current will flow in the direction such that it produces a magnetic field that opposes the change in the external magnetic field. This means that the current will flow in the negative direction of the external magnetic field at t = 2.0s, which is in the direction of -k hat.

 

[nlightner]

I am happy to help you with this question about electricity and magnetism. First, let's review some important concepts. Electricity and magnetism are two fundamental forces of nature that are closely related. Electricity is the flow of electric charge, while magnetism is the force exerted by moving electric charges. When an electric current flows through a wire, it creates a magnetic field around the wire. Similarly, a changing magnetic field can induce an electric current in a wire.

Now, let's apply these concepts to the given scenario. We have a circular wire loop with a radius of 1m, and it has a resistance of 5.0 ohms. This means that when a current flows through the loop, there will be a voltage drop of 5.0 volts across the loop. Next, we are told that there is a spatially uniform but time-dependent magnetic field throughout the region. This means that the strength and direction of the magnetic field are the same at all points in space, but it varies with time. The magnetic field is given by B = 1.0 T sin(wt) [k hat], where B is the magnetic field vector, T is time, and w is the angular frequency.

To find the magnetic flux passing through the loop at t = 2.0s, we need to use the formula for magnetic flux, which is Φ = BAcosθ, where Φ is the magnetic flux, B is the magnetic field, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop. In this case, the loop lies in the x-y plane, so the normal to the loop is in the z-direction. Therefore, the angle between the magnetic field and the normal is 90 degrees, and cos90 = 0. Plugging in the values, we get Φ = (1.0 T)(π1m2)(0) = 0 Wb.

Now, for the direction of the current induced in the loop at t = 2.0s, we can use Faraday's law of induction, which states that the induced voltage in a wire loop is proportional to the rate of change of magnetic flux through the loop. In other words, a changing magnetic field induces an electric current in the loop. In this case, the magnetic field is changing with time, so an electric current will be induced in the loop. The