Can a magnetic field be induced without an electric field? Can a magnetic field be induced without an electric field?
Because, as far as I know, a time varying electric field induces a magnetic field an vice versa. 
But in the case of conductors carrying currennt, it doesn't seem that electric field varies with time, then how is a magnetic field induced?
 A: One of Maxwell’s four equations for electromagnetism in a vacuum shows how magnetic fields are produced:
$$\nabla\times\mathbf{B}=\frac{1}{c}\left(4\pi\mathbf{J}+\frac{\partial\mathbf{E}}{\partial t}\right).$$
(I’ve written it in Gaussian units.)
From this equation you can see that there are two different sources for magnetic fields: the first is a current density, and the second is a changing electric field.
So to have a magnetic field you do not need to have a time-varying electric field. You can just have moving charge. But when a magnetic field is produced by moving charge, physicists don’t call it “induced”.
A: From Griffiths, Electrodynamics, Jefimenko’s equations are given as
$${\bf E}({\bf r},t) = \frac{1}{4 \pi \epsilon_0} \int [ \frac{\rho ({\bf r}',t_r)}{{\mathfrak r}^2} {\bf \hat{\mathfrak r}} + \frac{\dot{\rho} ({\bf r}',t_r)}{c {\mathfrak r}} {\bf \hat{\mathfrak r}} - \frac{{\bf {\dot J}} ({\bf r}',t_r)}{c^2 {\mathfrak r}}] d \tau',$$
$${\bf B}({\bf r},t) = \frac{\mu_0}{4 \pi} \int [\frac{{\bf {J}} ({\bf r}',t_r)}{{\mathfrak r}^2} + \frac{{\bf {\dot J}} ({\bf r}',t_r)}{c {\mathfrak r}} ] \times {\bf \hat{\mathfrak r}} d \tau'.$$
These equations show that to create a magnetic field you require either a steady current or/as well a changing current. If the current density is steady (so that ${\bf {\dot J}} \equiv 0$) then you can see that you can arrange for no electric field by having the charge density $\rho$ vanish everywhere. Another way to create a magnetic field is to have a time varying current density, which necessarily creates an electric field.
