Induced Electric Field work done

In summary: The magnetic field creates a vortex around the particle. In summary, a charged particle at a point P in a cylindrical region of radius R experiences a time varying magnetic field B. The work done by an external agent on the charged particle is independent of d and can be found by integrating E*x*cos(180-θ)=-kπR2.
  • #1
zorro
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Homework Statement



In a cylindrical region of radius R, there exists a time varying magnetic field B such that dB/dt=k(>0) . A charged particle having charge q is placed at the point P at a distance d (> R) from its centre O. Now, the particle is moved in the direction perpendicular to OP (see figure) by an external agent upto infinity so that there is no gain in kinetic energy of the charged particle. Show that the work done by the external agent is independent of d and find it.

attachment.php?attachmentid=34003&stc=1&d=1301911890.jpg

The Attempt at a Solution



Let the angle between the line joining the particle to the point O make an angle θ with the vertical.

E.dl=-kπR2
E*x*cos(180-θ)=-kπR2, where x is horizontal distance of the particle from the initial position.
Substituting x for dtanθ and solving, I get W= qkπR2 which is incorrect.

The answer given is qkπR2/4
 

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  • #2
What's your calculated E? How come I see no integration over cycle when you are clearly using Maxwell's Law? That's a blatant error,sorry.
 
  • #3
E=kπR2/xcosθ
Which integration are you talking about?
 
  • #4
E*x*cos(180-θ)=-kπR2
Ambiguous since Maxwell's Law is to be applied only to fixed mathematical loops.

Anyways since the angle force vector makes with the displacement is varying you will have to integrate to get the work.
 
  • #5
Explain how you went from
Abdul Quadeer said:
E.dl=-kπR2
to
E*x*cos(180-θ)=-kπR2, where x is horizontal distance of the particle from the initial position.
As aim1732 has suggested, you're not evaluating the LHS correctly. You can use Faraday's law to calculate E as a function of the displacement r. Once you have that, you can then find the force on the charge as a function of the displacement r and integrate it to find the work done.
 
  • #6
Ambiguous since Maxwell's Law is to be applied only to fixed mathematical loops.

But the particle is not moving along a loop here. Sorry I don't understand your point.


Explain how you went from...

At any instant, let the particle be at P distant 'x' from the initial point. E at that point has the direction as shown.

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dl vector is directed along the path traveled by the particle. I just used the dot product of two vectors.
 

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  • #7
Faraday's Law says

[tex]\oint_{\partial S} \mathbf{E}\cdot d\mathbf{l} = -\frac{\partial \Phi_{B,S}}{\partial t}[/tex]

where S is a surface and ∂S is the closed boundary of that surface. The expression -kπR2 is equal to the rate of change of flux, but your calculation of the line integral is wrong. The path the particle travels has nothing to do with the boundary of S.
 
  • #8
Ok.
What is the correct expression?
 
  • #9
First separately derive the electric field at a distance r from the centre of vortex. Then write out the differential work in terms of r and angle b/w displacement and force. Eliminate angle and integrate under proper limits..

By the way the existence of a vortex has absolutely nothing to do with the whether there is a charged particle at that point or not.
 

1. What is an induced electric field?

An induced electric field is a type of electric field that is created when a changing magnetic field passes through a conductor or circuit. It is caused by the movement of electric charges, which creates a force that can affect other charged particles in the area.

2. How does an induced electric field work?

An induced electric field works by the principle of electromagnetic induction. When a changing magnetic field passes through a conductor, it creates a flow of electric charges, which in turn creates an electric field. This electric field can then interact with other charged particles in the vicinity, causing them to move or experience a force.

3. What factors affect the strength of an induced electric field?

The strength of an induced electric field depends on several factors, including the rate of change of the magnetic field, the distance between the conductor and the magnet, and the material and shape of the conductor. It also depends on the strength of the original magnetic field and the resistance of the conductor.

4. What is the relationship between induced electric field and work done?

The work done by an induced electric field is equal to the product of the induced electric field strength and the distance over which the force acts. This means that the greater the strength of the induced electric field and the longer the distance over which it acts, the more work will be done.

5. What are some practical applications of induced electric fields?

Induced electric fields have many practical applications, including power generation, electric motors, transformers, and electromagnetic sensors. They are also used in technologies such as wireless charging, magnetic levitation, and induction heating. Additionally, they play a crucial role in the functioning of many electronic devices, such as speakers, microphones, and generators.

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