I was taught that , in a electromagnetic radiation , changing electric field is produced due to oscillation of charge this changing electric field give rise to 'changing' magnetic field and this changing magnetic field again gives rise to changing electric field . And this process is continue to occur
But my question is : a changing electric field give rise to constant magnetic field it doesn't produced changing magnetic field
So , how all of these will be correct?
If there is a doubt on the behaviour of the magnetic field in electromagnetic radiation, the best is to check it out with Maxwell's free equations:
$$\nabla \times\mathbf{E} =-\frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}$$
$$\nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t}$$
$$\nabla\cdot \mathbf{E} =0\quad \text{and} \quad \nabla\cdot \mathbf{B}=0$$
We start with a changing electrical field:
$$\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\cdot \mathbf{r} - \omega t)}$$
If we plug that in the first equation we get:
$$i \mathbf{k}\times \mathbf{E}_0 e^{i(\mathbf{k}\cdot \mathbf{r} - \omega t)} = -\frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}$$
We could now integrate over $t$ in order to get the result for $\mathbf{B}$. But we just guess the form of $\mathbf{B}$:
$$\mathbf{B} = \mathbf{B}_0 e^{i(\mathbf{k}\cdot \mathbf{r} - \omega t)}$$
Upon plugging that in the precedent equation we get:
$$i \mathbf{k}\times \mathbf{E}_0 e^{i(\mathbf{k}\cdot \mathbf{r} - \omega t)} = i\frac{\omega}{c} \mathbf{B}_0 e^{i(\mathbf{k}\cdot \mathbf{r} - \omega t)} \tag{1}$$
We learn the following from this result:
We have dispersion equation $|\mathbf{k}| = \frac{\omega}{c}$
The magnetic field vector is orthogonal to the (direction of the) wave vector and the electric field vector. Using $\nabla \cdot \mathbf{E} =0$ we also find that the wave vector $\mathbf{k}$ is orthogonal to the electric field vector.
$$\mathbf{k}\cdot \mathbf{E} =0$$
We can check our guess by evaluating the second Maxwell equation:
$$\nabla \times \mathbf{B} = -\frac{\omega}{c} \mathbf{E}_0 e^{i(\mathbf{k}\cdot \mathbf{r} - \omega t)}$$
and find that our guess $\mathbf{B} = \mathbf{B}_0 e^{i(\mathbf{k}\cdot \mathbf{r} - \omega t)}$
also fulfills the second Maxwell equation by using the dispersion relation $|\mathbf{k}|= \frac{\omega}{c}$:
$$i\mathbf{k}\times \mathbf{B}_0 e^{i(\mathbf{k}\cdot \mathbf{r} - \omega t)} = - i\frac{\omega}{c} \mathbf{E}_0 e^{i(\mathbf{k}\cdot \mathbf{r} - \omega t)} \tag{2}$$
We carry out a last check: We turn out second result (2) into the first one (1). For this we vector multiply the last equation with $\mathbf{k}$ and swap rhs with lhs:
$$-\mathbf{k}\times \mathbf{E_0} \frac{\omega}{c} = \mathbf{k}\times (\mathbf{k} \times \mathbf{B}_0) = (\mathbf{k}\cdot \mathbf{B}_0)\mathbf{k} - k^2 \mathbf{B}_0 = -k^2\mathbf{B}_0 $$
Because of $\nabla\cdot \mathbf{B}=0$ we know that $\mathbf{k}\cdot \mathbf{B}_0 =0$. Furthermore we know that $k^2 =(\frac{\omega}{c})^2$
So cancelling out a $-\frac{\omega}{c}$ we get our first result (1).
So we find that the the time and position dependence of the magnetic field is the same as for the electrical field, but the amplitude of both is the same in absolute values ( $|\mathbf{E}_0| =|\mathbf{B}_0|$. Note that the calculation was done in cgs-units where magnetic and electric field have the same units).
One can make the proof even more general in considering the fields (used here as check for their behaviour) as Fourier components of a more general behaviour on time and position. But this does not change the general picture, a varying electric field drives a varying magnetic field which again drives a varying electric field.
a changing electric field give rise to constant magnetic field
Only if derivative of the E-field is constant, what is a very special case, for a linear increasing or decreasing field. In general the derivative itself changes with time, what generates a changing magnetic field.
Not all of this is correct. This is correct: "changing electric field give rise to 'changing' magnetic field". And this is wrong according to Maxwell's equations. "a changing electric field give rise to constant magnetic field".
