 # Displacement Current

• The magnetic field due to current carrying conductor, ic is determined by using the Ampere’s circuit law.

\tt \int \overrightarrow{B} \cdot \overrightarrow{dl} = \mu_{0} i_{c}

• Conduction current, ic is produced by the time varying magnetic field \tt i_{c} \propto \frac{dB}{dt} \ and \ i_{c} = \frac{dq}{dt}
• The rate of change of electrical flux produces a current called DISPLACEMENT CURRENT, id.
• Flux due to electric field. φE = EA Cos θ
• \tt \overline{E} \cdot \overline{A} = \int \overline{E} \cdot \overline{ds}
• \tt i_{d} = \varepsilon_{0} \frac{d \phi_{E}}{dt}
• \tt i_{d} = \varepsilon_{0} A \frac{dE}{dt} {φ = E.A}
• When a variable electrical field is applied to the gap.

\tt i_{d} = A \varepsilon_{0} \frac{dE}{dt}                                     id = Displacement Current

• When a variable potential difference is applied to the plates of a condenser C then \tt i_{d} = C \frac{dv}{dt}
• \tt i_{c} = i_{d} \Rightarrow i_{c} = \frac{V}{X_{c}} = V \omega C
where, ic = conduction current,  id = displacement current
• Amperes circuit law \tt \int \overrightarrow{B} \cdot \overrightarrow{dl} = \mu_{0} \left(i_{c} + i_{d}\right)

= \tt \mu_{0} \left(i_{c} + \varepsilon_{0} \frac{d \phi_{E}}{dt}\right)

• \tt \int B \cdot dl = \mu_{0}i_{c} + \mu_{0} \varepsilon_{0} \frac{d \phi_{E}}{dt}
• \tt B = \frac{\mu_{0}}{2 \pi} i_{d} \frac{r}{R^{2}}
• The magnetic field at a distance R from the axis \tt B = \frac{\mu_{0}}{2 \pi} \frac{i_{d}}{R}. This is the maximum value.
• GAUSS LAW FOR ELECTRICITY \tt \int \overrightarrow{E} . \overrightarrow{dA} = q_{net}/ \varepsilon_{0}
• GAUSS LAW FOR MAGNETISM \tt \int \overrightarrow{B} . \overrightarrow{dA} = 0
\tt \int \overrightarrow{E} . \overrightarrow{dl} = - \frac{d \phi_{B}}{dt}
• \tt \int \overrightarrow{B} . \overrightarrow{dl} = \mu_{0}i_{c} + \mu_{0} \varepsilon_{0} \frac{d \phi_{E}}{dt} = \mu_{0} \left(i_{c} + i_{d}\right)
• The force acting on a charge, of moving in electric and magnetic fields which are similar to EM wave are existing simultaneously is \tt \overline{F} = q \left[ \overline{E} + \overline{V} \times \overline{B}\right]. This force is Lorentz force.

### View the Topic in this video From 00:34 To 57:46

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1. According to Ampere's circuital law, the magnetic field B is related to steady current I as
\oint_{c} \overrightarrow{B}\cdot \overrightarrow{dl} = \mu_{0}I

2. The sum of the conduction current and the displacement current. The generalised law is \oint B· dI\mu_{0}i_{c} + \mu_{0} \varepsilon_{0}\frac{d \phi_{E}}{dt}

3. \oint E·dA = Q/ε0 (Gauss's Law for electricity)

4. \oint B·dA = 0 (Gauss's Law for Magnetism)

5. \oint E·dl = \frac{-d \phi_{B}}{dt}(Faraday's Law)