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Transcript Chapter-3(phy-2)

(Electric Potential)
Electric Potential: The electrical state for which flow of charge
between two charged bodies takes place is called electric potential.
In other word, the electric or electrostatic potential at any point of
an electric field is defined as potential energy per unit charge at
that point. Electric potential is represented by letter V. That is,
V=U/q' or U=q'V…………………(1)
Electric potential is a scalar quantity since both charge and potential
energy are scalar quantities. S.I. unit of electric potential is Volt
which is equal to Joule per Coulomb.
In other word, The amount of work done in bringing a unit positive
charge from infinity to a point in electric field is called electric
potential at that point.
(Electric Potential)
If W is the work done in bringing a positive charge q from
infinity to a point in an electric field, then potential at that
point, according to definition is V = W/q. Potential V has a
sign. Potential at a point can be positive or negative. If the
electric field is due to positive charge, i.e. if the field is
positive, then in order to bring a unit positive charge from
infinity work is to be done against the repulsive force of the
similar charges. So, potential due to a positive charge is
positive. But if the field is due to negative charge, no work is
to be done by external force to bring a unit positive charge
from infinity. The attractive force will act between this positive
and negative charge. So, potential due to a negative charge is
(Electric Potential)
Potential difference: The difference of potentials
at two points in an electric field is called potential
difference. Or, The amount of work done in
transferring a unit positive charge from one point to
another point in an electric field is called potential
difference between the two points.
Let the potentials at two points A and B in an electric
field is VA and VB respectively. Then
VB - VA = ∆V = WAB/q0 = W/q0.
(Electric Potential)
Potential and Electric Field Strength:
Fig-1 shows two points A and B in
a uniform electric field E. Let the
distance of A from B is d. We
assume that a positive test charge
qo is being moved by an external
agent from A to B. The electric
force on the charge is qoE and
points downward. To move the
charge upward , the force on qo
must be countered by an external
force F of the same magnitude but
directed upward.
(Electric Potential)
The work done by the agent that supplies this force
is WAB = F × d = qoEd……………..(1)
If the electric potential difference between the points
A and B is VB - VA, then we can write,
VB - VA = WAB/q0 = qoEd/ q0= Ed…………….(2)
Eqn. (2) gives the relation connecting potential
difference and field strength for a simple special
case. From eqn. (2) it appears that another unit for E
is volt per meter. It can be proved that volt per meter
is identical with Newton per coulomb, or, 1 V/m = 1
(Electric Potential)
In Fig-1 B is at a higher potential than A because external agent
would have to do positive work to push a positive charge from A to
B against the direction of the field.
Let us now investigate the relation between V and E in the case in
which the field is not uniform and in which the test body moves
along a path which is not straight. The electric field exerts a force
qoE on the test charge as shown in Fig-2. To keep the test charge
from accelerating i.e., if the test charge is to move with a constant
velocity, the external agent must apply a force F = -qoE for all
positions of the test charge.
If the external agent causes the test charge to move through a
displacement dl along a path from A to B, the element of work done
by the external agent is Fcosθ dl = F . dl, where Fcosθ is the
component of the force in the direction of displacement.
(Electric Potential)
To find the total work WAB done by the external agent in
moving the test charge from A to B, we integrate (add
up) the work contributions for all infinitesimal segments
into which the path is divided. This lead to
(Electric Potential)
If the point A is at infinite distance then the potential VA
at infinity is taken as zero. Then, eqn. (4) gives the
potential V at the point B. By dropping the subscript B,
Eqns. (4) and (5) allow us to calculate either the
potential difference between ant two points or the
potential at any point if E is known at various points in
the field.
(Electric Potential)
Example-1: Let a test charge be moved without acceleration from
A to B over the path as shown in Fig-2. Compute the potential
difference between A and B.
(Electric Potential)
(Electric Potential)
Electric potential due to a point charge: Fig-3 shows two
points A and B near an isolated point charge q. For simplicity we
assume that A, B and q lie on a straight line. To compute the
potential difference between points A and B, we assume that
attest charge qo is moved without acceleration along a radial line
from A to B. The potential difference between the points A and B
is given by,
(Electric Potential)
(Electric Potential)
As we move a distance dl to the
left, we are moving in the direction
of decreasing r because r is
measured from q as an origin.
Thus, dl = - dr. Therefore, E . dl =
- E dl = E dr. Putting the value of E
. dl in eqn. (1), we get,
But we know,
where r is the distance
of the point from the
charge q.
Using this in eqn. (2),
we have,
(Electric Potential)
Let the reference point A be at infinity (i.e., rA→ ∞). Then
VA = 0 at this position, and dropping the subscript B, we
(Electric Potential)
Electric potential due to collection of charges:
For a collection of charges, the potential at a point is
calculated due to each individual charge, as if the other
charges were not present. These potentials are then
added and we obtain,
Where r1 is the distance of the point from the charge q1, r2
from q2 and r3 from q3, etc. Thus,
(Electric Potential)
If the charge distribution is
continuous, rather than
being a collection of points,
the sum in eqn. (2) must be
replaced by an integral. That
Where dq is a differential
element of the charge
distribution, r is its
distance from the point
at which V is to be
calculated and dV is the
potential it establishes at
that point.
(Electric Potential)
Example-2: What is the potential at the center of the
square if Fig-4?
(Electric Potential)
Example-3: What must the magnitude of an isolated
positive point charge be for the electric potential at 10
cm from the charge to be +100 Volts?
(Electric Potential)
Example-4: Three charges are placed at three corners
of a square as shown in Fig-5. Find the potential at
point P.
(Electric Potential)
Electric potential due to electric dipole
Two equal charges q of
opposite sign, separated by a
distance 2a, constitute an
electric dipole. The electric
dipole moment p has the
magnitude 2aq and points
from negative charge to the
positive charge. Fig-6 shows
such a dipole.
(Electric Potential)
Here we would like to derive an expression for
the electric potential V at any point of space due
to a dipole, provided only that the point is not too
close to the dipole. Let the point P where we
would like to calculate the potential. We know,
(Electric Potential)
Using eqn. (2) in eqn. (1), we get
(Electric Potential)
Eqn. (3) gives the exact value of the potential at the point
P. But for an ideal dipole (2a << r) r1 →r, r2 →r and ,
and in the limit,
The quantity 2aq is called the electric dipole moment, p.
Hence eqn. (4) reduces to,
(Electric Potential)
Special cases: (1) when the point P lies on the axial
line of the dipole on the side of the positive charge,
(2) when the point P lies on the axial line of the dipole
on the side of the negative charge,
(Electric Potential)
(3) when the point P lies on the equatorial line
of the dipole, θ = 900 so, cosθ = 0, so from eqn.
(5), VP = 0…………..(8).
This equation reflects the fact that no work is
done in bringing a charge from infinity to the
dipole along the perpendicular bisector of the
(Electric Potential)
Electric potential energy: Fig-7 shows two charges
q1 and q2 at a distance r apart. If the separation
between them is increased, work must be done by an
external agent. The work will be positive if the charges
are opposite in sign and negative otherwise. The
energy represented by this work can be thought of as
stored in the system q1 + q2 as electric potential
energy. Like all forms of potential energy, this energy
can also be transformed into other forms.
(Electric Potential)
The electric potential energy of a system of point charges
may be defined as the work required to assemble this
system of charges by bringing them in from an infinite
distance. We assume that the charges are all at rest when
they are infinitely separated, that is they have no initial
kinetic energy.
In Fig-7 let us imagine q2 removed to infinity and at rest.
The electric potential at the original site of q2 due to q1 is
given by,
(Electric Potential)
If q2 is now moved from infinity to its original distance r from q1,
the work required is, W = Vq2
[As, V = W/q]
This work is precisely the electric potential energy U of
the system q1 + q2. Thus,
The subscript of r emphasizes that the distance
involved is in between point charges q1 and q2.
(Electric Potential)
Example-5: Two protons in a nucleus of U238 are 6.0×
10-15 m apart. What is their mutual electric potential
(Electric Potential)
Example-6: Three charges are arranged as in the figure. What is
their mutual potential energy? Assume that q = 1.0 × 10-7 C and a
= 10cm.
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