Above is a connection of half-wave dipole antenna. The book states that the current profile is same at any given time instant on both the radiators, each of quarter wavelength length.
Now my confusion is clearly the coaxial cable is connected to both radiators. Say I am sending the signal through centre wire and the braid of the coaxial cable is always at ground. Now at any given time instant, only left end of the cable is going to have non zero potential, while the right end is always at $0V$. Then how come both radiators would have same current profile at any given time when the excitation is different for left and right radiators!
To be very specific, say at any given time instant the voltage at the left end is 2.3V, so standing wave is generated in the left radiator, but on the right radiator the voltage is zero (braid is at ground), so how any excitation would happen on the right radiator.
Support will be greatly appreciated. I am a total newbie in antenna systems.