EM-2.4-1

Work & Electrostatic Energy

Consider: A stationary configuration of source charges is used to generate a electric field E(r). A test charge QT is moved from point a to point b in this electric field.

Question:How much mechanical work W is done on the test charge QTin moving it (slowly) from point a to point b?

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Work & Electrostatic Energy (conti.)

The mechanical force required to balance the electrostatic force is

At any point r along the path a → b, the electrostatic force acting on QTis

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Work & Electrostatic Energy (conti.)

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Work & Electrostatic Energy (conti.)

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Example 1

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Example 1 (conti.)

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Example 1 (conti.)

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Example 1 (conti.)

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Example 2

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Example 2 (conti.)

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Example 2 (conti.)

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Work done around a cloed path

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ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION

How much work does it take to assemble a collection of point charges – bringing them in from infinity, one by one?

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ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.)

Bringing in the first charge q1 –NO work (W1 =0), since there is no electric field present, initially.

Bringing in 2nd charge q2 -

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ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.)

Bringing in 3rd charge

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ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.)

Bringing in the 4th charge

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ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.)

The total work necessary to assemble the first 4 charges is

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ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.)

For N charges

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CONTINUOUS CHARGE DISTRIBUTION

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CONTINUOUS CHARGE DISTRIBUTION (conti.)

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CONTINUOUS CHARGE DISTRIBUTION (conti.)

Using Gauss’ Law:

Using Divergence theorem (eq. 1.59, p. 37)

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CONTINUOUS CHARGE DISTRIBUTION (conti.)

Or

Thus

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CONTINUOUS CHARGE DISTRIBUTION (conti.)

For a single point charge

Why?

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Electrostatic energy density

~ valid for continuous charge distributions

Define electrostatic energy density

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Electrostatic energy density (conti.)

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ELECTROSTATIC ENERGY AND THE SUPERPOSITION PRINCIPLE

Potential energy is quadratic in the electric field

Therefore, if we double E, i.e., E->2E, potential energy in the electrostatic field quadruples (4x).

Thus, work done to assemble the charge distribution does NOT obey the principle of linear superposition

EM-2.4-27

ELECTROSTATIC ENERGY AND THE SUPERPOSITION PRINCIPLE (conti.)