Question

An electromagnetic field is said to be conservative when curl of the field is ......

Answer 1

A conservative vector field is one which can be written as the gradient of a scalar function. That is, if V=V(r)V=V(r) is a conservative vector field, then there is some function f=f(r)f=f(r) such that

(1)  V=∇f=x^∂f∂x+y^∂f∂y+z^∂f∂zV=∇f=x^∂f∂x+y^∂f∂y+z^∂f∂z

Now, the curl of  VV is defined as

(2)  ∇×V=x^(∂Vz∂y−∂Vy∂z)+y^(∂Vx∂z−∂Vz∂x)∇×V=x^(∂Vz∂y−∂Vy∂z)+y^(∂Vx∂z−∂Vz∂x)+z^(∂Vy∂x−∂Vx∂y)+z^(∂Vy∂x−∂Vx∂y) 

Plugging (1) into (2), we find

(3)  ∇×V=x^(∂2f∂y∂z−∂2f∂z∂y)∇×V=x^(∂2f∂y∂z−∂2f∂z∂y)+y^(∂2f∂z∂x−∂2f∂x∂z)+y^(∂2f∂z∂x−∂2f∂x∂z)+z^(∂2f∂x∂y−∂2f∂y∂x)+z^(∂2f∂x∂y−∂2f∂y∂x)

However, Clairaut's theorem asserts that second derivatives are symmetric with respect to the order of differentiation,

∂2f(x1,x2)∂x1∂x2=∂2f(x1,x2)∂x2∂x1∂2f(x1,x2)∂x1∂x2=∂2f(x1,x2)∂x2∂x1

So, you can see that each term in parenthesis in (3) is going to be zero. Hence, the curl of any conservative vector field VV is zero,

∇×V=∇×∇f=0