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Displacement Current | What is Displacement Current | Displacement Current Formula

Displacement Current | What is Displacement Current | Displacement Current Formula :- Under the magnetic effect of electric current, we know that the relationship between the generated magnetic field and the electric current is shown by Ampere’s circuital law as follows :-

$\displaystyle \oint{\overrightarrow{B.}\overrightarrow{dl}}={{\mu }_{0}}I$

But when this rule is applied while charging and discharging of a capacitor then this rule appears inconsistent.

Suppose a parallel plate capacitor is being charged by connecting it to a battery. During the process of charging, the charging current flowing through the connecting wire changes with time.

Due to the charging current a magnetic field is generated around the connecting wire. During the process of charging (and discharging) the electric field generated between the plates of the capacitor changes, but after the capacitor is fully charged, there is no change in the electric field between the two plates.

In the above figure, to find the magnetic field at point P, a circular amperian loop has been made passing through point P. To use Ampere’s circular law, if a plane surface perpendicular to the wire S1 is chosen, then

$\displaystyle \oint\limits_{{{S}_{1}}}{\overrightarrow{B.}\overrightarrow{dl}}={{\mu }_{0}}I(t)\text{ }.....(1)$

Here I(t) is the charging current or conduction current.

Now if the bulging surface S₂ is chosen (since in Ampere’s circuital law the surface can be chosen at random, this does not affect the result), then

$\displaystyle \oint\limits_{{{S}_{2}}}{\overrightarrow{B.}\overrightarrow{dl}}={{\mu }_{0}}\times 0=0\text{ }.....(2)$

In the above equation (2), the value of current has been taken to be zero because no current appears to flow through the bulged out surface S₂ (i.e. between the plates of the capacitor).

Since both the surfaces S₁ and S₂ are located very close, hence

$\displaystyle \oint\limits_{{{S}_{1}}}{\overrightarrow{B.}\overrightarrow{dl}}=\oint\limits_{{{S}_{2}}}{\overrightarrow{B.}\overrightarrow{dl}}\text{ }.....(3)$

Now if calculated in one way then the magnetic field at point P has some non-zero value [Equation (1)] and if calculated in another way then the magnetic field at point P is zero [Equation (2)]. Therefore, equations (1), (2) and (3) cannot be valid together. Thus Ampere’s circuital law appears inconsistent.

Because this contradiction arises due to Ampere’s circular law, there is probably a term missing in this law. This omitted term should be such that no matter which surface we use (flat surface S₁ or bulged out surface S₂) we want to get the same value of magnetic field at point P.

If we look carefully at the diagram of charging a capacitor, this contradiction can be resolved. Is there a change in the value of any quantity passing through the surface between the plates of the capacitor? Yes, actually the electric field (E) is changing between them. Therefore, to overcome the contradiction, Maxwell imagined that not only the conduction current flowing through the conductor generates magnetic field, but the magnetic field can also be generated due to the changing electric field passing through vacuum or any dielectric medium.

Therefore, it can be assumed that a changing electric field is equivalent to an electric current. This electric current can be considered present as long as there is a change in the electric field. Due to this electric current, the same type of magnetic field is generated as is generated due to conduction current flowing through a conductor. This current is called Maxwell’s displacement current.

Displacement Current | What is Displacement Current | Displacement Current Formula

Definition

The electric current which arises due to change in electric field or electric flux is called displacement current.

Displacement Current | What is Displacement Current | Displacement Current Formula

Formula (expression) of displacement current

Suppose the area of the plates of the capacitor is A and the charge on its plates at any time is q, then the magnitude of the electric field (E) between the plates will be :

$\displaystyle E=\frac{\sigma }{{{\varepsilon }_{0}}}=\frac{q}{{{\varepsilon }_{0}}A}$

According to Gauss’s law changing electric flux (Φ_E) between the plates,

$\displaystyle {{\phi }_{E}}=EA=\frac{q}{{{\varepsilon }_{0}}A}.A=\frac{q}{{{\varepsilon }_{0}}}$

$\displaystyle \Rightarrow q={{\varepsilon }_{0}}{{\phi }_{E}}$

Now since the charge q on the plates of the capacitor is changing with time, the displacement current

$\displaystyle {{I}_{d}}=\frac{dq}{dt}=\frac{d}{dt}({{\varepsilon }_{0}}{{\phi }_{E}})={{\varepsilon }_{0}}\frac{d{{\phi }_{E}}}{dt}$

Hence, Maxwell’s formula for displacement current,

$\displaystyle {{I}_{d}}={{\varepsilon }_{0}}\frac{d{{\phi }_{E}}}{dt}\text{ }.....(4)$

Displacement Current | What is Displacement Current | Displacement Current Formula

Modified Ampere’s circuital law

Based on the above discussion, according to Maxwell there are two sources of magnetic field : –

(i) Charging current or conduction current flowing through the connecting wires and

(ii) Displacement current generated due to changing electric field (E) or changing electric flux (Φ_E) between the plates of the capacitor.

Therefore, the modified Ampere’s circuital law is as follows:

$\displaystyle \oint{\overrightarrow{B.}\overrightarrow{dl}}={{\mu }_{0}}({{I}_{c}}+{{I}_{d}})$

$\displaystyle \oint{\overrightarrow{B.}\overrightarrow{dl}}={{\mu }_{0}}({{I}_{c}}+{{\varepsilon }_{0}}\frac{d{{\phi }_{E}}}{dt})\text{ }.....(5)$

The above equation is called Ampere-Maxwell’s circuital law.

Conclusion

The induced emf according to Faraday’s law of electromagnetic induction,

$\displaystyle e=-\frac{d{{\phi }_{B}}}{dt}\text{ }.....(6)$

Hence, time varying magnetic field generates electric field.

From Ampere-Maxwell’s circuital law,

$\displaystyle \oint{\overrightarrow{B.}\overrightarrow{dl}}={{\mu }_{0}}({{I}_{c}}+{{\varepsilon }_{0}}\frac{d{{\phi }_{E}}}{dt})\text{ }.....(7)$

According to the above formula, time varying electric field generates magnetic field.

Thus, from equations (6) and (7), time varying magnetic field generates time varying electric field and time varying electric field generates time varying magnetic field. Therefore, a wave is generated in which the changes in the electric field and the magnetic field are the sources of each other’s origin. This wave is called electromagnetic wave.

Displacement Current | What is Displacement Current | Displacement Current Formula

Displacement Current | What is Displacement Current | Displacement Current Formula

Definition

Displacement Current | What is Displacement Current | Displacement Current Formula

Formula (expression) of displacement current

Displacement Current | What is Displacement Current | Displacement Current Formula

Modified Ampere’s circuital law

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