Induced Current Density On Disk Due to Changing Magnetic Field

In summary, the problem involves a thin conducting disc of radius a and conductivity \sigma in the x-y plane with a spatially uniform induction given by B=B_0 cos\(\omega t \)\hat{z}. The goal is to find the induced current density \vec{J} in the disc using the equations \vec{J} = \sigma ( \vec{E} + \vec{v} \times \vec{B} ) and \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}. The solution involves finding \vec{
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Homework Statement



A very thin conducting disc of radius [tex]a[/tex] and conductivity [tex]\sigma[/tex] lies in the x-y plane with the origin at its center. A spatially uniform induction is present and given by [tex]B=B_0 cos\(\omega t \)\hat{z}[/tex]. Find the induced current density [tex]\vec{J}[/tex] in the disc.

Homework Equations



[tex] \vec{J} = \sigma ( \vec{E} + \vec{v} \times \vec{B} ) [/tex]
[tex] \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} [/tex]
[tex] \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}[/tex]

The Attempt at a Solution



I'm utterly confused on this problem, I really just don't know where to start. The first equation for [tex]\vec{J}[/tex] is utterly useless as our medium is stationary. The second is also not too helpful as it doesn't take into account the magnetic field's time dependence. The third also suffers from this problem. So that is my first hiccup. I know it's zero progress but this subject is hard for me :(
 
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  • #2
I'm silly, I think I got it started:

[tex] \vec{\nabla} \times \vec{E} = - \frac{ \partial \vec{B} }{\partial t} = + B_0 \omega \sin(\omega t) \hat{z} [/tex]
[tex] \oint_C \vec{E} \cdot d\vec{s} = \oint_C E_\phi \hat{\phi} \cdot \rho d\phi \hat{\phi} = 2 \pi \rho E_\phi [/tex]
[tex] = \int_S (\vec{\nabla} \times \vec{E}) \cdot d\vec{a} = \omega B_0 \sin(\omega t) \int da_z [/tex]
[tex] E_\phi = \frac{1}{2} \omega B_0 \sin(\omega t) [/tex]
 

1. What is induced current density on a disk?

Induced current density on a disk refers to the flow of electric current that is generated in a conducting disk when it is exposed to a changing magnetic field. This phenomenon is known as electromagnetic induction, and it is a fundamental principle in physics.

2. How is induced current density calculated?

The induced current density on a disk can be calculated using the formula: J = σBωsin(ωt), where J is the current density, σ is the conductivity of the disk, B is the strength of the magnetic field, ω is the angular frequency of the changing magnetic field, and t is time. This formula is derived from Faraday's law of induction.

3. What factors affect the induced current density on a disk?

The induced current density on a disk is affected by several factors, including the strength and direction of the changing magnetic field, the conductivity of the disk material, and the frequency of the changing magnetic field. Additionally, the size and shape of the disk can also impact the induced current density.

4. How does induced current density affect the disk?

Induced current density on a disk can have several effects, depending on the specific application. In some cases, it can cause the disk to heat up due to the flow of electric current. In other cases, it can be used to power electrical devices or generate magnetic fields for various purposes.

5. What are some real-life applications of induced current density on a disk?

Induced current density on a disk has several practical applications, such as in electric generators, transformers, and induction cooktops. It is also used in various medical devices, such as MRI machines, to produce images of the body. Induced current density is also utilized in wireless charging technology for electronic devices.

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