To find the frequency at which the conduction current ((I_c)) and displacement current ((I_d)) become equal in a material, we first need to understand the equations governing these currents in relation to the material’s properties.

The conduction current density ((J_c)) is given by Ohm’s law:

[J_c = sigma E]

where (sigma) is the conductivity of the material and (E) is the electric field.

The displacement current density ((J_d)) can be described in relation to the changing electric field as:

[J_d = epsilon frac{dE}{dt}]

where (epsilon) is the permittivity of the material, and (frac{dE}{dt}) represents the rate of change of the electric field with respect to time.

Given that (sigma = 1) S/m (unity conductivity) and (epsilon = 2), to find the frequency ((f)) at which (J_c = J_d), set the expressions for (J_c) and (J_d) equal to each other and solve for (f).

[J_c = J_d]

[sigma E = epsilon frac{dE}{dt}]

Because we’re concerned with frequencies, a useful form of (E) to consider is one that oscillates sinusoidally, so let (E = E_0 sin(

To calculate the displacement current density ((J_d)) when the electric flux density ((D)) is given by (D = 20sin(0.5t)), we’ll make use of Maxwell’s displacement current concept. The displacement current density can be found using the equation:

[

J_d = frac{dD}{dt}

]

Given that (D = 20sin(0.5t)), let’s differentiate it with respect to (t):

[

frac{dD}{dt} = 20cos(0.5t) times 0.5 = 10cos(0.5t)

]

So, the displacement current density, (J_d), is:

[

J_d = 10cos(0.5t)

]

This expression provides the displacement current density as a function of time.

To find the velocity of an electron when its kinetic energy (K) is equal to 1 electron volt (eV), we use the relation between kinetic energy and velocity, and the given values for the charge (e) and mass (m) of an electron.

The kinetic energy K in joules for an electron can be given by the equation ( K = frac{1}{2} mv^2 ), where (m) is the mass of the electron and (v) is its velocity.

Given:

– (K = 1 , text{eV} = 1.6 times 10^{-19} , text{J}) (since (1 , text{eV} = 1.6 times 10^{-19} , text{J}))

– (m = 9.1 times 10^{-31} , text{kg})

– (e = 1.6 times 10^{-19} , text{C}) (charge of the electron, although it’s not directly used in the calculation for velocity, it’s useful for understanding the energy conversion)

Starting with the relationship between kinetic energy and velocity:

[ K = frac{1}{2} mv^2 ]

Solving for (v):

[ v = sqrt{frac{2K}{m}} ]

Plugging in the values:

[ v =

The superconducting materials will be independent of external magnetic field strength up to a critical limit. Beyond this critical magnetic field strength, superconductivity is destroyed. Superconductivity is also independent of normal resistive mechanisms present in conventional conductors but depends on factors like temperature, magnetic field, and material purity.