GenE123
Jayaram, S
ELECTRIC POTENTIAL, MAGNETIC FIELDS AND
FARADAY’S LAW
(Refer to your Physics text, Chapters 24, 28 and 30)
Chapter 24
24.2 Electric Potential Energy
24.3 Electric Potential (Voltage)
24.5 Calculating the Potential from the Field
24.6 Potential due to a Point Charge
24.7 Potential due to a Group of Point Charges
Chapter 28
28.2 The Magnetic Field
→
28.3 Definition of B
28.8 Magnetic Force on a Current Carrying Wire
Chapter 30
30.2 Some Early Experiments
30.3 Faraday’s Law of Induction
30.4 Lenz’s Law
GenE123
Jayaram, S
Electric Potential
24.2 Electric Potential Energy
From study of mechanics, we know that when we lift an object of mass m , a
vertical distance h near the earth’s surface, the work done goes into potential
energy.
P ⋅ E = mgh
Where, g the acceleration due to gravity
If the mass is allowed to fall down, the stored potential energy is converted into
kinetic energy. (K.E)
K ⋅E =
1
mv 2
2
Where, v is the velocity
The force associated in the above earth-mass system is the gravitational force.
Like the gravitational force, the electrostatic force (often called the Coulomb
force) is also conservative.
Thus, an electric potential energy, U can be assigned to a system where an
electrostatic force acts between two or more charged particles. Since charges
exert force on each other, energy must be expended (or work must be done) in
moving a charge from its initial position to a final position.
∴ The resulting change in the potential energy of the system ∆U is given as:
∆U
= Uf − Ui = −W
24.3 Electric Potential
Although the potential energy of a particle depends on the charge, the quantity
U/q, defined as electric potential, V has a unique value at any point in an electric
field. Thus the electric potential, potential energy per unit charge, V,
U
q
Where, U is the final potential energy, with Ui set to zero and U = −W∞
V=
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Jayaram, S
W∞ is the work done by the electric field on a charged particle, as the particle
moves in from infinity to a point f .
The electric potential difference ∆V , between any two points i and f in an electric
field, is equal to the difference in potential energy per unit charge between the
two points:
Moving a charge q in an electric field, E
∆V = Vf − Vi =
Uf Ui ∆U
−
=
q
q
q
∆U = Change in energy = -W
∴ ∆V = −
W
q
The potential difference between two points is thus the negative of the work done
by the force, to move a unit charge from one point to the other.
∆V can be positive, negative or zero, depending on the sign and magnitudes of
q and W.
Unlike electric field, electric potential is a scalar quantity.
Unit for potential (voltage, V) is joule per coulomb (J/C).
1 (J/C) = 1 volt (V), and the unit volt is given the symbol V.
Hence the unit for electric field can be visualized as, volt/meter (V/m)
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Jayaram, S
The electric field, E has its original unit (N/C). We can show that this unit is same
as V/m.
(N / C ) =
N ⋅ m  Joule(J )   1 
1
=
⋅   ⋅ = (Volt ) ⋅

C ⋅m 
C
m
 m
=> (V/m)
24.5 Calculating the Potential from the Field
First, find the work done on a positive test charge by the field along the path
shown (from i to f)
r r
The differential work done dW = F .d s
But,
→
→
F = q0 E
r r
dW
=
q
E
.d s
Hence,
0
Total work done on the particle by the field in moving it along (i) to (f) is,
f →
→
W = q0 ∫ E ⋅ ds
i
By definition of electric potential difference,
(Vf
f →
→
− Vi ) = − ∫ E ⋅ ds
i
GenE123
Jayaram, S
Therefore, the potential difference between any two points (i) and (f) in an electric
field is the negative of the line integral (line integral means integral along a path)
→
→
of E ⋅ ds from (i) to (f).
Since the electrostatic force is conservative, all paths yield the same results “Path Independence”. (See example below)
Or,
ur →
V = − ∫ E ⋅ ds
f
i
Above equation represents the potential V at any point (f) in the electric field
relative to the zero potential at point (i). ≡> This is equivalent to the potential
energy per unit charge at a point in an electric field.
Note: The subscript f is dropped onVf .
Example 1
In the following figure, find the potential difference Vf − Vi by moving a
positive test charge q0 , from i to f along the paths shown.
(a)
(b)
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Jayaram, S
24.6 Potential due to a Point Charge:
→
We know that the electric field E at any point P can be obtained from,
E =
q
4πε 0 r 2
But,
∞
∆V = Vf − Vi = − ∫ E ⋅ dr
R
→
→
ur uur
(Here, E ⋅ ds = Eds cosθ , with ds in the same direction as E )
→
In our geometry, since E is directed away (radially), with θ =0 and cosθ =1,
ur uur
E ⋅ ds = E ⋅ dr
Setting Vf = 0 at infinity (away for q), andVi = V at the point P, which is at a
distance R from q ,
∞
q
R
4πε 0 r 2
Vf − Vi = ( 0 − V ) = − ∫
−V = −
q
∞
1
dr =
∫
4πε 0 R r 2
V =
q
4πε 0R
dr
∞
q  1
4πε 0  r  R
GenE123
Jayaram, S
The general formula to express the potential V, as the electric potential due to a
particle of charge q at any radial distance r from the particle:
V=
q
4πε 0 r
A positively charged particle produces a positive electric potential, and a negatively charged particle produces a negative
potential.
24.7 Potential due to a Group of Point Charges:
If we have more than one point charge, we can find the net potential at a point
due to all charges using the superposition principle.
∴ For n charges,
n
V = ∑Vi =
i =1
1
4πε 0
n
qi
i =1
i
∑r
Where,
qi is the value of the i th charge
ri is the radial distance of the given point from i th charge
→
Unlike E the sum is an algebraic sum and not a vector sum
Example 2
Consider a point charge q = 1.0 µC, point A at a distance d1 = 2.0 m from q
and point B at distance d2 = 1.0 m. What is the electric potential if the points
A and B are located as shown in (a) and (b)?
(a)
B ------- d2 ------- +------------------- d1------------------ A
(b)
B
|
|
d2
|
|
+------------------ d1------------------ A
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Jayaram, S
Magnetic Fields
→
The existence of a magnetic field B , at some point in space can be demonstrated
using a compass needle.
•
•
If the direction of the needle is in the direction of the magnetic field of the
earth, then it indicates that there are no other magnetic fields around the
compass.
If the direction is different, it indicates the presence of any nearby objects
like magnets or objects that carry electric currents.
Note: Earth is a large magnet.
In the above figure, N – North Pole and S – South Pole
Unlike electric charge, there are no “monopoles” i.e. magnetic poles always exist in pairs.
Magnetic field exists around permanent magnets and around any current
carrying elements.
Don’t forget that the moving electric charge causes the current flow.
dq
= i (t ) ⇒ Rate of change of charge = current.
dt
GenE123
Jayaram, S
Further, experimentally it has also been observed that when a charged particle
moves through a magnetic field, a force due to the magnetic field can act on the
particle. This force is different from the electrostatic force.
→
Quantitatively, the magnetic field B can be defined as follows:
→
→
→
FB = q v X B
(Cross Product)
Where,
→
→
F = force on the particle due to the existence of B (magnetic field)
rB
v = velocity of the moving charge.
Magnitude of the force,
FB = q vB sin φ
→
→
Where, φ is the angle between the directions of velocity v and magnetic field B .
→
→
Illustration of use of right-hand-rule to find the direction for (v X B )
Simply use the right-hand rule for vector products:
→
•
Stretch your fingers in the direction of the first vector, in this case v
•
Bend the fingers into the direction of the second vector, in this case B
→
→
→
Then the thumb indicates the direction of the product, in this case ( v X B ).
→
→
→
If q is positive, then the direction of F is same as that of ( v X B ).
B
→
→
→
If q is negative, the F direction is opposite to that of ( v X B ).
→
→
B
→
The force F , v and B are perpendicular to each other.
B
The above equation for force tells us many things:
•
The force is zero if q = 0 or if q is stationary.
•
The force is zero if v and B are parallel ( φ = 00 ) or anti parallel ( φ = 180o )
•
The force is maximum when v and B are perpendicular to each other (of
course for q ≠ 0 ).
→
→
→
→
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Jayaram, S
→
The SI unit for B is tesla (T).
1 tesla = 1 T = 1
Newton
 N 
= 1

(Coulomb )(meter / sec ond )
 A⋅m 
( Since Coulumb / second = Ampere )
28.8 Magnetic force on a Current - Carrying Wire:
Significance of dot and cross notations
Let,
L = length of the wire (for that matter any current carrying element)
i = a constant current that is flowing through the wire (in this case, upwards)
The charge that passes a certain distance in a time t = L/v is
q = it = i
∴ F8 = qvB sin φ =
L
v
iL
vB sin90o
v
Or,
FB = iLB
GenE123
Jayaram, S
The above equation represents the magnetic force that acts on a length L of
→
straight wire, carrying a current i and immersed in a magnetic field B that is
perpendicular to the wire.
If the magnetic field is not perpendicular to the wire,
→
→
FB = i L X B
→
Note: L is a vector and not the current i.
→
L represents the length of the current carrying wire in the direction of the current
flow.
FB = iLB sin φ
Where,
→
→
→
φ is the angle between L and B . The direction of FB is same as that
→ →
of the cross product  L x B  .


We will work on some numerical problems in class.
Faraday’s Law of Induction
Faraday’s Law:
We already rknow
ur that a current carrying wire in a magnetic field experiences a
force, FB = iL × B . The converse is also true. If a wire (conductor) is moved in a
magnetic field, a current flows through the wire.
Faraday first observed the above facts
•
The current appears only when there exists a relative motion between the
moving wire and the magnetic field.
•
If the wire is moved faster, higher current flows.
•
The direction, in which the current flows in an external circuit, depends on
the direction in which the magnetic field is directed relative to the wire.
GenE123
Jayaram, S
The law states “An emf is induced in the loop, when the number of magnetic field
lines that pass through the loop change”
To define Faraday’s law in mathematical terms, we define the amount of
magnetic field lines that pass through a loop (cross section) as:
→
→
Magnetic Flux through Area A, φB = ∫ B ⋅ dA
Where,
→
dA is a vector of magnitude dA
→
→
B⋅ dA = B ⋅ dA cosθ = BdA , when the loop lies in a plane which is perpendicular to
→
the magnetic field, B .
∴ φB = BA
Unit for φB is weber = 1 tesla m2
1 T m2 = 1 Wb
The induced emf can now be defined in terms of flux as: “the magnitude of the
emf (E) induced in a conducting loop, is equal to the rate at which the magnetic
flux ( φB ) through that loop changes with time.”
E=
−
dφ B
dt
=> Faraday’s Law
If magnetic flux passes through N number of turns in the loop,
E= −N
dφ B
dt
The direction of the induced current is found using Lenz’s Law.
The Lenz’s law states that “An induced current has a direction, such that the
magnetic field due to the current opposes the change in the magnetic flux that
induces the current”.