ELECTROMAGNETICS THEORY SEE 2523 ELECTROSTATIC FIELD ELECTRIC FLUX

ELECTROMAGNETICS THEORY (SEE 2523) ELECTROSTATIC FIELD

ELECTRIC FLUX DENSITY, GAUSS’S LAW & DIVERGENCE THEOREM

INTRODUCTION ¨ Gauss’s law is used to find electric flux intensity. ¨ Using Gauss’s law the work turns out to be easier but it only convenient for symmetrical charges distribution. ¨ Electric flux density, any charge distribution. gives a total flux lines obtained by

3. 1: ELECTRIC FLUX ¨ Electric flux is the number of electric field lines penetrating a surface or an area. ¨ The lines of the electric flux emanate from +Q and terminates on –Q. ¨ The lines of the electric flux is in same direction with electric field, . ¨ In SI unit, is used to represent the flux lines, (1)

¨ Consider two point charges, +Q and –Q as shown in Fig. 3. 1

3. 2: ELECTRIC FLUX DENSITY ¨ Defined as total electric flux lines emit from a surface (2) ¨ The electric flux density, with is a vector in same direction , as shown in Fig. 3. 2. The flux lines directly perpendicular to the surface Δs. Fig. 3. 2

¨ From (2) (3) ¨ Consider a point charge Q is located at the origin in a free space. Assume a sphere with a radius, r centered at the origin shown in Fig. 3. 3 (4) V/m (5) Fig. 3. 3 ¨ From (4) & (5): C/m 2 (6)

3. 3: GAUSS’S LAW ¨ Consider the charge distribution, Q surrounded by a closed surface in Fig. 3. 4. ¨ From the concept of the electric flux , then (7) S Fig. 3. 4

Gauss’s law states that the total electric flux through any closed surface is equal to the total charge enclosed by that surface S (8) Fig. 3. 4

¨ If the surface consist of ρv, then : ¨ This law is true for symmetrical charge distributions only.

3. 4 DIVERGENCE ¨ Consider a vector in a coordinate system as shown in Fig. 3. 5

¨ In order to compute the total flux out of this volume, we need to do the surface integral at all the 6 surfaces : ¨The results computed to yield :

¨ Let the volume dv approach zero then applied Gauss’s law. => First Maxwell’s equation for Electrostatics field The net outward flux from a closed surface as the volume shrinks to zero.

(a) (b) (c) Fig. 3. 6 (a) positive divergence (b) negative divergence (c) zero divergence Divergence is applied with vector quantity but yield a scalar result.

3. 5: DIVERGENCE THEOREM ¨ Starting with the Gauss’s law : and Using Hence : The divergence theorem states that the total outward flux of a vector field through the closed surface is the same as the volume integral of the divergence.