The x-component of the electric field categorized by an electric potential V(x,y,z) be capable of be expressed as -dV/dx holding y and z constant. This would propose that the units of E would be the units of V divided by the units of x that is in SI units volts per meter. Are the units of E volts per meter? If not what is incorrect with the argument above?

Ans:

dV is an minuscule electrical potential difference. It perhaps infinitesimal but the difference between two values of electric potential has the same units as a single value of electric potential namely volts whether it is infinitesimal or not. A similar case is able to be made for the infinitesimal difference in position dx. The amount dx is a distance infinitesimal though it may be. Like such it have units of distance. Note that the units of electric field are certainly N/C but this does not rule out V/m as units of electric field. In reality a volt is a J/C and since a joule is a Nm a volt is as well a Nm/C. Restore the V (volts) with Nm/C in the units V/m (volts per meter) yields (Nm/C)/m which is equivalent to N/C the SI units that we usually think of the electric field as having. In summary a V/m is a N/C so the argument that E cannot have units of V/m because it has units of N/C is clearly wrong.