0195136047.pdf

8 CIRCUIT CONCEPTS

Loop 1 Loop 2

R

I 1 I^2

a 12

dl 1

a 21 dl 2

Figure 1.1.2Illustration of Ampere’s law (of

force).

2. The magnitude of the force is inversely proportional to the square of the distance between
   the current elements.


3. To determine the direction of, say, the force acting on the current elementI 1 dl ̄ 1 , thecross
   productdl ̄ 2 × ̄a 21 must be found. Then crossingd ̄l 1 with the resulting vector will yield
   the direction ofdF ̄ 21.


4. Each current element is acted upon by amagnetic fielddue to the other current element,

dF ̄ 21 =I 1 dl ̄ 1 ×B ̄ 2 (1.1.8a)

dF ̄ 12 =I 2 d ̄l 2 ×B ̄ 1 (1.1.8b)

whereB ̄is known as themagnetic flux density vectorwith units of N/A·m, commonly

known as webers per square meter (Wb/m^2 ) or tesla (T).

Current distribution is the source of magnetic field, just as charge distribution is the source

of electric field. As a consequence of Equations (1.1.7) and (1.1.8), it can be seen that

B ̄ 2 =μ^0

4 π

I 2 d ̄l 2 × ̄a 21 (1.1.9a)

B ̄ 1 =μ^0

4 π

I 1 dl ̄ 1 × ̄a 12

R^2

(1.1.9b)

which depend on the medium parameter. Equation (1.1.9) is known as theBiot–Savart law.

Equation (1.1.8) can be expressed in terms of moving charge, since current is due to the flow

of charges. WithI=dq/dtandd ̄l= ̄vdt, wherev ̄is the velocity, Equation (1.1.8) can be

rewritten as

dF ̄=

(

dq

dt

)

(vdt) ̄ ×B ̄=dq (v ̄×B) ̄ (1.1.10)

Thus it follows that the forceF ̄experienced by a test chargeqmoving with a velocityv ̄in a

magnetic field of flux densityB ̄is given by

F ̄=q(v ̄×B) ̄ (1.1.11)

The expression for the total force acting on a test chargeqmoving with velocityv ̄in a region

characterized by electric field intensityE ̄and a magnetic field of flux densityB ̄is

F ̄=F ̄E+F ̄M=q(E ̄+ ̄v×B) ̄ (1.1.12)

which is known as theLorentz force equation.