Show that Faraday's Law holds

In summary, the conversation discusses the calculation of the negative time derivative of B(r, t) and the difficulty with calculating the Curl of the electric field. The speaker also asks for help in showing the relationship between the two calculations and asks for assistance in a warm-up exercise.
  • #1
Blanchdog
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22
Homework Statement
Suppose that an electric field is given by ##E(r, t) = E_0 \text{cos}(k \cdot r - \omega t + \phi) ##, where ##k \perp E_0## and ##\phi## is a constant phase. Show that
$$B(r, t) = \frac{k~\text{x}~E_0}{\omega} \text{cos}((k \cdot r - \omega t + \phi)$$ is consistent with Faraday's Law.
Relevant Equations
Faradays Law: $$\nabla~\text x~E = -\frac{\partial B}{\partial t}$$
I've calculated the negative time derivative of B(r, t) as: $$-\frac{\partial B}{\partial t} = k~\text x~E_0~\text{sin}(k \cdot r - \omega t + \phi)$$ The cross product can be easily expanded, I'd just rather not do the LaTeX for if I can avoid it.

The Curl of the electric field (##\nabla~\text{x}~ E##) is giving more trouble though. I should end up with a sine wave (or a cosine offset by pi/2) but as best I can tell the curl doesn't affect the stuff within the cosine at all since k and r are dotted together into a scalar. How do I show that the curl of the electric field is equal to the result of the negative time derivative of of the magnetic field above, or did I make a mistake in calculating that derivative?
 
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  • #2
Blanchdog said:
as best I can tell the curl doesn't affect the stuff within the cosine at all
As a warmup exercise, calculate ##\frac{\partial}{\partial x} (\mathbf k \cdot \mathbf r)##.
 
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  • #3
Blanchdog said:
I've calculated the negative time derivative of B(r, t) as: $$-\frac{\partial B}{\partial t} = k~\text x~E_0~\text{sin}(k \cdot r - \omega t + \phi)$$
Check the signs
 
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  • #4
That was very helpful, I was able to solve it once I fixed the sign of the sine and saw the pattern of the derivatives I was able to do after your warm up.
 
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1. How does Faraday's Law relate to electromagnetic induction?

Faraday's Law states that a changing magnetic field will induce an electromotive force (EMF) in a conductor. This means that when a conductor is moved through a magnetic field or when the magnetic field changes, an electric current will be produced in the conductor. This phenomenon is known as electromagnetic induction.

2. What is the mathematical equation for Faraday's Law?

The mathematical equation for Faraday's Law is: EMF = -N(dΦ/dt), where EMF is the electromotive force, N is the number of turns in the conductor, and dΦ/dt is the rate of change of the magnetic flux through the conductor. This equation shows the relationship between the induced EMF and the changing magnetic field.

3. How is Faraday's Law used in practical applications?

Faraday's Law is used in many practical applications, such as generators, transformers, and electric motors. Generators use Faraday's Law to convert mechanical energy into electrical energy, while transformers use it to change the voltage of an alternating current. Electric motors use Faraday's Law to convert electrical energy into mechanical energy.

4. What are the factors that affect the magnitude of the induced EMF?

The magnitude of the induced EMF is affected by several factors, including the strength of the magnetic field, the velocity of the conductor, and the angle between the magnetic field and the conductor. Additionally, the number of turns in the conductor and the rate of change of the magnetic field also affect the magnitude of the induced EMF.

5. How is Faraday's Law related to Lenz's Law?

Lenz's Law is a consequence of Faraday's Law. It states that the direction of the induced current in a conductor will always be such that it opposes the change that produced it. This means that the induced current will create a magnetic field that opposes the change in the original magnetic field, according to Faraday's Law.

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