Show that the retarded scalar potential given by
and the retarded vector potential given by
,
satisfy the Lorentz gauge condition.
To show that the retarded scalar potential \(\phi(\vec{r},t)\) and the retarded vector potential \(\vec{A}(\vec{r},t)\) satisfy the Lorentz gauge condition, we need to verify that: \[ \nabla \cdot \vec{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0 \] --- ### Step 1: Compute \( \nabla \cdot \vec{A} \) The retarded vector potential is given by: \[ \vec{A}(\vec{r},t) = \frac{\mu_0}{4\pi} \int \frac{\vec{j}(\vec{r}', t - \frac{|\vec{r} - \vec{r}'|}{c})}{|\vec{r} - \vec{r}'|} dV' \] Taking the divergence: \[ \nabla \cdot \vec{A} = \frac{\mu_0}{4\pi} \int \nabla \cdot \left( \frac{\vec{j}(\vec{r}', t - \frac{|\vec{r} - \vec{r}'|}{c})}{|\vec{r} ___ _____ ______ ___ ____ _______ ______ ___.
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Obtain expressions for electric and magnetic fields in terms of scalar and vector potentials using homogeneous Maxwell's equations.
A circular wire loop of radius ris kept in the x-y plane in a region of uniform magnetic field pointing in the z-direction. The magnitude of the magnetic field is held constant at B=0 from time t=0 to time.t=t1 Thereafter, the magnetic field is increased at constant rate from B=0at time t=t2 to BB till time f-12 and then reduced at a constant rate from B= Bo to B=0 at time t=t3. a) Compute the magnetic flux through the circular wire loop as a function of time. b) Compute the induced emf in the circular loop as a function of time. c) Plot the magnetic flux and induced emf as a function of time.
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Explain the phenomenon of time dilation and obtain expression relating the time intervals between two event observed by different inertial observers.
Show that the retarded scalar potential given by
and the retarded vector potential given by
,
satisfy the Lorentz gauge condition.
A plane electromagnetic wave propagating in z-direction is incident on the boundary between two linear dielectric media. The boundary is in the x-y plane perpendicular to the direction of propagation of the wave. Using the boundary conditions on the electric and magnetic fields associated with the wave, show that if the incident wave is plane polarised then the reflected and transmitted waves also have the same polarisation.