Show what the magnitude of induced emf

In summary, induced emf is a phenomenon in which a changing magnetic field causes an electric current to flow in a conductor. It is caused by a changing magnetic field and can be calculated using Faraday's Law of Induction. The magnitude of induced emf is affected by factors such as the strength and rate of change of the magnetic field, as well as the number of turns and material of the conductor. Some applications of induced emf include electric generators, transformers, and scientific instruments such as electromagnetic sensors and particle accelerators.
  • #1
auk411
57
0
Someone who knows what they are talking about: Show what the magnitude of induced emf

Homework Statement



Consider a magnetic field B = K(x3z2,0, -x2z3)sinωt in the region of interest, where K and ω are positive constants and t is variable time. Show that the magnitude of the induced emf around a circle R in the plane z = a with its center at x = 0, y = 0, z = a is:
ε = (K/4)∏a3R4ωcosωt

Homework Equations



Fluxb = ∫B . dA

The Attempt at a Solution



Since the normal vector points in the k direction, we only have to worry about Bz.

∫Bzdydx. So -∫∫(sinwt)x2a3dydx.

The make the change to polar:

-aK3∫∫(sinwt)(rcosθ)2r dr dθ = -(K/4)a3R4∫cosθsin(wt) dθ.

This doesn't get me anywhere. I'm not really sure what I'm supposed to be integrating over, which is probably why I'm stuck.
 
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  • #2
Hi auk411! :smile:

Let's start with Bz.
Can you write it down separately?
It seems you did not copy it correctly.

Secondly you did not bring the constant out of the integral properly.

Furthermore in polar coordinates you would integrate r from 0 to R, and theta from 0 to 2pi.

And for the tip: rewrite (cosθ)2 using cos2θ.
 
  • #3
I like Serena said:
Hi auk411! :smile:

Let's start with Bz.
Can you write it down separately?
It seems you did not copy it correctly.

Secondly you did not bring the constant out of the integral properly.

Furthermore in polar coordinates you would integrate r from 0 to R, and theta from 0 to 2pi.

And for the tip: rewrite (cosθ)2 using cos2θ.

(cosθ)2 using cos2θ.[/QUOTE] ... huh, what trig identity are you using.

this still doesn't answer the most pressing question. t varies, theta varies and r varies. we have 3 varying variables in a DOUBLE integral. I see no way to reduce them to two. How do I get around this.
 
  • #4
First things first.
You appear to have skipped my question, so I'll answer it myself:

Bz = -Kx2a3sinωt

This is not what you used.

auk411 said:
(cosθ)2 using cos2θ. ... huh, what trig identity are you using.

cos(2θ) = 2 cos2θ - 1


auk411 said:
this still doesn't answer the most pressing question. t varies, theta varies and r varies. we have 3 varying variables in a DOUBLE integral. I see no way to reduce them to two. How do I get around this.

No, you have 2 variables.
You appear to be thinking spherical coordinates, but you should be thinking in cylindrical coordinates.

z is constant at z=a.
Only the other 2 vary.
 
  • #5


I would approach this problem by first understanding the concept of induced emf. Induced emf is the electromotive force that is generated in a conductor when it is exposed to a changing magnetic field. This force is caused by the movement of free electrons in the conductor, which creates a current.

In this problem, we are given a magnetic field B that is varying with time and we need to find the magnitude of the induced emf in a circular loop at a specific location. To do this, we can use Faraday's law of induction, which states that the induced emf in a closed loop is equal to the negative rate of change of magnetic flux through the loop.

In mathematical terms, this can be written as:

ε = -dΦ/dt

Where ε is the induced emf, Φ is the magnetic flux, and t is time.

In order to calculate the magnetic flux, we need to first determine the magnetic field at the location of the circular loop. We are given the magnetic field B in terms of x, y, and z coordinates, so we can use the equation B = ∇ x A, where A is the vector potential.

In this case, A = (0, x3z2, -x2z3)sinωt, so B = (0, 3x2z2sinωt, -2xz3sinωt).

Next, we need to find the area vector for the circular loop. Since the loop is in the z = a plane, the area vector is in the k direction, and its magnitude is equal to the area of the circle, which is πR2. Therefore, the area vector is given by A = (0, 0, πR2).

Now, we can calculate the magnetic flux through the circular loop as:

Φ = ∫B . dA = ∫(0, 3x2z2sinωt, -2xz3sinωt) . (0, 0, πR2) dA

= ∫-2xz3sinωt dA

= -2sinωt ∫x(πR2)z3 dA

= -2sinωt ∫x(πR2)z3 rdrdθ

= -2sinωt (πR2) ∫∫xrcosθrdrdθ

=
 

What is induced emf?

Induced emf stands for induced electromotive force, and it is a phenomenon in which a changing magnetic field causes an electric current to flow in a conductor. This can occur through various means such as moving a magnet near a conductor or changing the current in a nearby circuit.

What causes induced emf?

Induced emf is caused by a changing magnetic field. When a conductor is exposed to a changing magnetic field, it creates a force that can push electrons to flow in a certain direction, resulting in an induced emf. This is known as Faraday's Law of Induction.

How is the magnitude of induced emf calculated?

The magnitude of induced emf can be calculated using the formula E = -N(dΦ/dt), where E is the induced emf, N is the number of turns in the conductor, and dΦ/dt is the rate of change of magnetic flux through the conductor. This formula is also known as Faraday's Law.

What factors affect the magnitude of induced emf?

The magnitude of induced emf is affected by a few factors such as the strength of the magnetic field, the rate of change of the magnetic field, and the number of turns in the conductor. Additionally, the material of the conductor and its dimensions can also affect the magnitude of induced emf.

What are some applications of induced emf?

Induced emf has various applications in our daily lives, such as in electric generators, transformers, and induction cooktops. It is also used in many scientific instruments, such as electromagnetic sensors and particle accelerators.

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