Question

I want the answers done neatly with explanations!

Problem 1) Find the magnetic field B (value and direction) such that the voltage across the capacitor in the following circuit is 5V. Is this a practical way to charge the capacitor? Explain fully.

Answer 1

we all know that the capacitor is nothing but the two parallel plate which get charged accordingly when a voltage or potential difference is applied with them, .This plate is is always given a specified amount of the thickness as well as the area of the cross section to deal with it . The capacitance between the two plate of the capacitor is measured by the following formula :

C=($\epsilon *A$)/d

where C= capacitance

$\epsilon$= permeability  of the medium

d= distance between the plates

in between the two plates there is no charge difference present so the current density between the two plates is equal to the zero. so according to the maxwell equation

E= q(t)/($\epsilon$*a)

where

E= instantaneous electric field

q(t) = charge present within the circuit instantaneously

a= area of the cross-section

$\epsilon _{0}\mu _{0}\frac{\partial E}{\partial t}=(\mu _{0}/a)\frac{\partial q}{\partial t}$

so the differential version of the Ampère–Maxwell law in the gap is

curl B =

(Ho/a)

[ AMPERE MAXWELL LAW :∇ × H = Jc + Jd

the ampere Maxwell law is the law mixture of both the ampere law and the Maxwell law , this law gives the clear picture about how the above equation has come into the front as this equation actually shows the reality of the things and from the magnetic field theory we know that the

B=$\mu _{0}$ H ]

[ for the vector analysis chapter we know that the curl a= line integral of vector a]

$\oint B.dl=(\mu _{0}/a)\frac{\partial q}{\partial t}\int e.ds$

considering it as the circular surface we get

B.2$\pi$r =

at

B=$\mu _{0}/a\frac{\partial q}{\partial t} e. r/(2*A)$

THE MAGNETIC FIELD WILL BE PERPENDICULAR TO THE DIRECTION of the applied voltage and the charging direction .

one thing should be made very clearly to the mind that the magnetic field is not only due to the constants of the Maxwell's equation but also due to the the charging phenomenon of the capacitor . Till now no reasons have been found to prove that the capacitor have no magnetic field associated with it .