Displacement Current Density Calculation

To calculate the displacement current density, we need to use the equation:

Jd = ε * dE/dt

where Jd is the displacement current density, ε is the permittivity of free space, and dE/dt is the rate of change of electric flux density.

Given that the electric flux density is 20sin(0.5t), we can differentiate it with respect to time to find dE/dt.

Differentiating Electric Flux Density

dE/dt = d/dt (20sin(0.5t))

Using the chain rule of differentiation, we can differentiate the sine function:

dE/dt = 10cos(0.5t) * d/dt (0.5t)

The derivative of 0.5t is 0.5, so we have:

dE/dt = 10cos(0.5t) * 0.5

Simplifying the expression, we get:

dE/dt = 5cos(0.5t)

Calculating Displacement Current Density

Now that we have the rate of change of electric flux density (dE/dt), we can calculate the displacement current density using the equation Jd = ε * dE/dt.

Given that ε is the permittivity of free space, which is a constant value, we can substitute the values:

Jd = ε * 5cos(0.5t)

Since the options given are in the form of a trigonometric function, we can rewrite the expression as:

Jd = 5εcos(0.5t)

Comparing this with the given options, we see that option 'B' is the correct answer:

Jd = 10cos(0.5t)

Conclusion

In conclusion, the displacement current density is calculated by differentiating the electric flux density with respect to time and then multiplying it by the permittivity of free space. In this case, the rate of change of electric flux density is 5cos(0.5t), leading to a displacement current density of 10cos(0.5t), which matches option 'B'.