Chapter 24

Gauss’s Law

In a tabletop plasma ball, the colorful lines emanating from the sphere give evidence of strong electric fields. Using Gauss’s law, we show in this chapter that the electric field surrounding a uniformly charged sphere is identical to that of a point charge. (Steve Cole/Getty Images)

Performance Criteria

Electric Flux

Area A

Electric flux

(N.m²/C)

Note that: Electric flux is proportional to the number of electric field lines penetrating some surface.

Electric field (N /C)

Area of the surface m²

(A) If the surface (A) is perpendicular to the field lines.

Note that:

1) The flux through a surface of fixed area A has a maximum value EA when the surface is perpendicular to the field (when the normal to the surface is parallel to the field, that is, when �θ = 0^o in Fig. 24.2).

2) The flux is zero when the surface is parallel to the field (when the normal to the surface is perpendicular to the field, that is, when θ = 90^o).

3) The product E cos θ is the component of the electric field perpendicular to the surface. Then,

where we use E_n as the component of the electric field normal to the surface.

(B) If the surface (A) is not perpendicular to the field (E)

(c) In more general situations, the electric field may vary over a large surface.

If the area of each element approaches zero, the number of elements approaches infinity and the sum is replaced by an integral.

Note that:

1) If the area element is a part of a closed surface, the direction of the area vector is chosen so that the vector points outward from the surface.

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2) If the area element is not part of a closed surface, the direction of the area vector is chosen so that the angle between the area vector and the electric field vector is less than or equal to 90°.

(D) The flux through a closed surface.

For elements such as (3) , where the field lines are crossing the surface from outside to inside, 180^o > θ>90^o and the flux is negative because cos θ is negative.

Gauss’s Law

The general relationship between the net electric flux through a closed surface (often called a gaussian surface) and the charge enclosed by the surface is known as Gauss’s law.

The electric field lines are directed radially outward and hence are perpendicular to the surface at every point on the surface.

The electric net flux

N.m²/C

We have moved E outside of the integral because, by symmetry, E is constant over the surface

Where as;

&;

Because the surface is spherical

Since,

Then,

Then,

And,

The electric net flux N.m²/C

The net electric flux

N.m²/C

Source Charge

C

Vacuum permittivity

Note that:

1) The net flux through the spherical surface is proportional to the charge inside the surface.

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2) The flux is independent of the radius r .

Because the area of the spherical surface is proportional to r², whereas the electric field is proportional to 1/r². Therefore, in the product of area and electric field, the dependence on r cancels.

Q2_ The flux is independent of the radius r. Explain why.

Example:

Consider several closed surfaces surrounding a charge q as shown in Figure 24.7. Surface S₁ is spherical, but surfaces S₂ and S₃ are not.

Just a reminder:

Note that:

Applying this result to Example 24.1, we see that the net flux through the cube is zero because there is no charge inside the cube.

The superposition principle states that the electric field due to many charges is the vector sum of the electric fields produced by the individual charges.

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Therefore, the flux through any closed surface can be expressed as;

The surface S surrounds only one charge, q₁; hence, the net flux through S is q₁/Є_o.. The flux through S due to charges q₂, q₃, and q₄ outside it is zero.

(because each electric field line from these charges that enters S at one point leaves it at another).

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The surface S’ surrounds charges q₂ and q₃; hence, the net flux through it is (q₂ + q₃)/ Є_o.

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Finally, the net flux through surface S’’ is zero (because there is no charge inside this surface).

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Charge q₄ does not contribute to the net flux through any of the surfaces.

Q3_Zero Flux Is Not Zero Field. Explain.

In two situations, there is zero flux through a closed surface, either:

(1) there are no charged particles enclosed by the surface or

(2) there are charged particles enclosed, but the net charge inside the surface is zero.

For either situation, it is incorrect to conclude that the electric field on the surface is zero. Gauss’s law states that the electric flux is proportional to the enclosed charge, not the electric field.

The mathematical form of Gauss’s law is a generalization of what we have just described and states that the net flux through any closed surface is:

Application of Gauss’s Law

to

Various Charge Distributions

Application of Gauss’s Law to Various Charge Distributions

Q4_Gaussian Surfaces Are Not Real. Explain.

A gaussian surface is an imaginary surface you construct to satisfy the conditions listed here. It does not have to coincide with a physical surface in the situation.

By symmetry, E has the same value everywhere on the surface, which satisfies condition (1), so we can remove E from the integral.

A good electrical conductor contains charges (electrons) that are not bound to any atom and therefore are free to move about within the material

Q5_What is main property of a good electric conductor?

When there is no net motion of charge within a conductor, the conductor is in electrostatic equilibrium.

… Electrostatic Equilibrium

Conductor in …

Q6_ What are the properties of conductor in electrostatic equilibrium?

Q7_The electric field is zero everywhere inside the conductor in electrostatic equilibrium, whether the conductor is solid or hollow. Explain why.

- Before the external field is applied, free electrons are uniformly distributed throughout the conductor.

- When the external field is applied, the free electrons accelerate to the left in Figure 24.16, causing a plane of negative charge to accumulate on the left surface. The movement of electrons to the left results in a plane of positive charge on the right surface.

- These planes of charge create an additional electric field inside the conductor that opposes the external field.

- Then, the electrons move, until it reach a point where the magnitude of the internal field equals that of the external field, resulting in a net field of zero inside the conductor, which makes it stop.

- If the conductor is hollow, the electric field inside the conductor is also zero, whether we consider points in the conductor or in the cavity within the conductor.

Q8_If the conductor in electrostatic equilibrium is isolated and carries a charge, the charge resides on its surface. Explain.

Figure 24.17 shows an arbitrarily shaped conductor.

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A gaussian surface is drawn inside the conductor and can be very close to the conductor’s surface.

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As we have just shown, the electric field everywhere inside the conductor is zero when it is in electrostatic equilibrium.

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Therefore, the electric field must be zero at every point on the gaussian surface, and the net flux through this gaussian surface is zero.

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From this result and Gauss’s law, we conclude that the net charge inside the gaussian surface is zero.

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Because there can be no net charge inside the gaussian surface (which is arbitrarily close to the conductor’s surface), any net charge on the conductor must reside on its surface.

Q10_How can we find the magnitude of the electric field?

- Hence, the net flux through the Gaussian surface is equal to that through only the flat face outside the conductor, where the field is perpendicular to the gaussian surface.

- Using conditions (1) and (2) for this face, the flux is EA, where E is the electric field just outside the conductor and A is the area of the cylinder’s face.

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- Applying Gauss’s law to this surface gives:

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