Field Theory EC 44

1
Field Theory EC 44

• 1.Review
• 2. Divergence Theorem

2
Electrostatics - a review

• We deal with stationary charges so that the
  electrostatic energy is conserved
• Two types of Charges ve and ve
• Electric lines originate from ve charges and
  terminate on ve chareges
• No of such lines threading a surface is called
  flux
• Each charge produces a flux equal to its
  magnitude.

3
Electrostatics - Review

• Gausss law
• Integral form
• Differential form
• Divergence theorem

4
Field Theory EC 44

• Energy and Potential

5
Energy in Electrostatic Field

• At any point in an electric field the electric
  field strength E is defined as the force
  experienced by a unit positive charge at that
  point.

?q is a small test charge
6
Energy in Electrostatic Field

• If we wish to move a test charge Q against the
  electric field, we have to exert a force equal to
  and opposite to that exerted by the field.
• This requires us to expend energy i.e., to do
  work against the field.
• If we wish to move the test charge in the
  direction of the field, the energy expended
  becomes negative i.e., we do not do work but the
  field does.

7
Energy in Electrostatic Field

• Suppose a charge Q is placed at a point inside an
  electric field E. Then it experiences a force due
  to the field given by

Next Suppose we wish to move the charge Q over a
distance
inside the field E
The component of this force in the direction of
which we have to overcome is,
8
Energy in Electrostatic Field

• Thus the force which we have to apply is equal
  and opposite to the force due to the field. i.e.,
  
• The expended energy, given by the product of
  force and distance(as in mechanics) i.e., the
  differential work done dW by the external agency
  in moving Q is

9
Energy in Electrostatic Field

• Therefore the total work done W i.e., the energy
  expended to move the charge a finite distance is

where the path must be specified before the
integral can be evaluated. The charge is assumed
to be stationary at both its initial and final
positions
10
Energy in Electrostatic Field

• Given the electric field
• Find the differential amount of work done in
  moving a 6 nC charge a distance 2µm starting at
  P(2,-2,3) and proceeding in the direction

Answer -149.33 fJ 149.33 fJ 0 fJ
11
Energy in Electrostatic Field

• The line integral
• Let us now choose a path and break it into
  sufficently large number of very small segments,
  multiply the component field along each segment
  and then add the results for all the segments.
  The integral obtained is exact only when the
  number of segments chosen become infinite.

12
Energy in Electrostatic Field

• 
  A Final Position
• 
  ?L 8 EL8 E
• 
  ?L 7 EL7
• 
  ?L 6 EL6
• ?L 5
  EL5
• ?L 4 EL4
• ?L 3 EL3
• ?L 2 EL2
• ?L 1 EL1
• B initial position
  E

13
Energy in Electrostatic Field

• In the figure we have chosen a path from an
  initial position B to a final position A in a
  uniform electric field. The path is divided into
  a number of segments ?L1, ?L2, ?L3, , ?L7. The
  components along each segment are EL1, EL2, EL3,
  , EL7. Then the work done in moving a charge
  from B to A is then,
• In vector notation, this equation is

14
Energy in Electrostatic Field
Or in integral form, In a uniform field,
15
Energy in Electrostatic Field

• Work out your self the examples 4.1 and 4.2
• Work out your self the drill problems D 4.1 D4.2
  and D 4.3.
• Also work out the example given in page 89,90 of
  the text Hayt Buck

16
Potential

• Potential at any point in an electric field is
  defined as the work done in moving a unit
  positive charge from infinity to that point.
• The work done by an external source in moving a
  charge Q from one point to another in an electric
  field E is

P
17
Potential

• Potential Difference V between two points in an
  electric field is the work done (by an external
  source) in moving a unit positive charge from one
  point to another.
• VAB signifies the pd between points A and B and
  is the work done in moving the unit charge from B
  (last named) to A (first named). Thus in
  determining VAB, B is the initial point B is
  often taken as infinity. The unit of potential
  difference is J/C or Volts

18
Potential

• Compute the work done in moving a unit positive
  charge in a radial path from ? a to ? b for a
  uniformly charged infinite line and hence find
  the p.d
• Find the p d between points A and B at a radial
  distance rA to rB from a point charge Q.
• (pages 89, 90 and 91 of the text)

19
Potential

• To find the p d between points A and B at radial
  distances rA and rB from a point charge Q. Let us
  choose origin at Q.

20
Potential

• If we let the point r rB recede to infinity,
  the potential at rA becomes absolute potential or
  simply potential as

Work out D4.5. Find the potential on the z axis
for a uniform line charge ?L in the form of a
ring of Radius a. Page 97 of the text.
21
Potential

• The expression for the potential difference shows
  that no work is done in carrying a unit charge
  around any closed path. i.e.,

Work out D 4.6
22
Potential

• It is often convenient to speak of the potential
  or absolute potential at any point in a field.
• i.e., we measure the p d with respect to a
  specific reference point, having a zero
  potential.
• Most universal zero reference point in ground.
• Another widely used reference point is
  infinity.

23
Potential

• If the potential at point A is VA and that at B
  is VB, then
• VAB VA - VB
• where VA and VB have the same zero reference
  point