282
XVII
FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS
So far we have merely repeated in precise form the results
of the preceding paragraph. We now go further and conclude
that a system of variable currents exerts electric forces of the
form (2). These may be represented in the form (3). Hence
unless they are constant they will give rise to magnetic forces
of form (4). And these must be added as a correction to the
known magnetic forces of form (1). To arrive at the expres-
sion of the forces (2) in the form (3) we put
-AU,
dt
- Adv
dt
- A ?dW₁
ƏR ƏQ
= A
Əz
=
Assuming for the present that
A
მყ
ƏP ƏR
ӘР
Əz
-
მე:
ƏQ ap
= A
dt
Əx Əy
ӘР. ƏQƏR
+ +
эх ду Əz
=
0
(a)
we get, by differentiating the second equation with respect to
z, the third with respect to y, and subtracting the results,
and thence
P:
=
-
A
dav awi
-
dt az ду
1 d/a
A
4π dt az
=
V2P,
-
a
W 1
We get similar expressions for Q and R. It is easy to see
that these satisfy the equation (a), and the assumption of the
truth of this equation is justified.
From the values of P, Q, R follow the magnetic forces
produced by their variation. The x-component is
-
=
-
dt
1
A³
4π
3
V₁
-
W
dt2 dz
1
მყ
This term we must add to the component L, of the previously
assumed magnetic force. Let us call the component thus