2.4-work And Energy
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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 QT in moving it (slowly) from point a to point b?
EM-2.4-1
Work & Electrostatic Energy (conti.) At any point r along the path a → b, The mechanical force required to balance the the electrostatic force acting on QT electrostatic force is is
EM-2.4-2
Work & Electrostatic Energy (conti.)
EM-2.4-3
Work & Electrostatic Energy (conti.)
EM-2.4-4
Example 1
EM-2.4-5
Example 1 (conti.)
EM-2.4-6
Example 1 (conti.)
EM-2.4-7
Example 1 (conti.)
EM-2.4-8
Example 2
EM-2.4-9
Example 2 (conti.)
EM-2.4-10
Example 2 (conti.)
EM-2.4-11
Work done around a cloed path
EM-2.4-12
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?
EM-2.4-13
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 -
EM-2.4-14
ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.) Bringing in 3rd charge
EM-2.4-15
ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.) Bringing in the 4th charge
EM-2.4-16
ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.) The total work necessary to assemble the first 4 charges is
EM-2.4-17
ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.) For N charges
EM-2.4-18
CONTINUOUS CHARGE DISTRIBUTION
EM-2.4-19
CONTINUOUS CHARGE DISTRIBUTION (conti.)
EM-2.4-20
CONTINUOUS CHARGE DISTRIBUTION (conti.) Using Gauss’ Law:
Using Divergence theorem (eq. 1.59, p. 37)
EM-2.4-21
CONTINUOUS CHARGE DISTRIBUTION (conti.) Or Thus
EM-2.4-22
CONTINUOUS CHARGE DISTRIBUTION (conti.)
For a single point charge
Why? EM-2.4-23
Electrostatic energy density
~ valid for continuous charge distributions
Define electrostatic energy density
EM-2.4-24
Electrostatic energy density (conti.)
EM-2.4-25
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-2.4-26
ELECTROSTATIC ENERGY AND THE SUPERPOSITION PRINCIPLE (conti.)
EM-2.4-27
EM-2.4-1
Work & Electrostatic Energy (conti.) At any point r along the path a → b, The mechanical force required to balance the the electrostatic force acting on QT electrostatic force is is
EM-2.4-2
Work & Electrostatic Energy (conti.)
EM-2.4-3
Work & Electrostatic Energy (conti.)
EM-2.4-4
Example 1
EM-2.4-5
Example 1 (conti.)
EM-2.4-6
Example 1 (conti.)
EM-2.4-7
Example 1 (conti.)
EM-2.4-8
Example 2
EM-2.4-9
Example 2 (conti.)
EM-2.4-10
Example 2 (conti.)
EM-2.4-11
Work done around a cloed path
EM-2.4-12
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?
EM-2.4-13
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 -
EM-2.4-14
ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.) Bringing in 3rd charge
EM-2.4-15
ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.) Bringing in the 4th charge
EM-2.4-16
ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.) The total work necessary to assemble the first 4 charges is
EM-2.4-17
ELECTROSTATIC ENERGY OF ASSEMBLY OF POINT CHARGES DISTRIBUTION (conti.) For N charges
EM-2.4-18
CONTINUOUS CHARGE DISTRIBUTION
EM-2.4-19
CONTINUOUS CHARGE DISTRIBUTION (conti.)
EM-2.4-20
CONTINUOUS CHARGE DISTRIBUTION (conti.) Using Gauss’ Law:
Using Divergence theorem (eq. 1.59, p. 37)
EM-2.4-21
CONTINUOUS CHARGE DISTRIBUTION (conti.) Or Thus
EM-2.4-22
CONTINUOUS CHARGE DISTRIBUTION (conti.)
For a single point charge
Why? EM-2.4-23
Electrostatic energy density
~ valid for continuous charge distributions
Define electrostatic energy density
EM-2.4-24
Electrostatic energy density (conti.)
EM-2.4-25
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-2.4-26
ELECTROSTATIC ENERGY AND THE SUPERPOSITION PRINCIPLE (conti.)
EM-2.4-27
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