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Question : 31
Total: 35
SECTION - D
(a) (i) Explain how free electrons in a metal at constant temperature attain an average velocity under the action of an electric field. Hence obtain an expression for it. (ii) Consider two conducting wires
OR
(b) (i) A cell emf of (E) and internal resistance (
(ii) Three cells, each of emf
Obtain expressions for (i) current flowing in the circuit, and (ii) the terminal potential difference across the equivalent cell.
Solution: 👈: Video Solution
(a) (i) In absence of any electric field, the free electrons in metals move haphazardly in all possible directions and hence, develop no net flow of current. When an electric field is applied, a force acts on the electrons and the electrons now tend to move in the direction of the force.
When electron collides with lattice, its velocity becomes instantaneously zero and then again it starts moving in a specific direction due to the applied electric field.
If the average time between two collisions (relaxation time) isτ
l =
a τ 2
Wherel =
a = acceleration =
E = electric field intensity
e = charge of electron
m = mass of electron
∴ l =
τ 2
Drift velocity =
= v d =
τ
= K
∴ v d = K × E
Thus, free electrons in a metal at constant temperature under the action of an electric field attain a constant average velocity.
(b) Since, the wires are joined in series current flowing through then will be same.
Let the current in bothA B
Diameter being same, there areas of cross section are also same. Let it beA
So, in wireA
I = n A e A v d A
In wireB
I = n B e A v d B
Taking the ratio,
1 =
×
Or, 1 = 1.5 ×
∴
=
= 2 : 3
OR
(b) (i) Terminal voltage vs. load resistance graph:
V = E − i r
Andi =
∴V = E −
Or,V = E
∴V =
Terminal voltage vs. current graph:
(ii) Three cells combination diagram is given below,
Current through cells( i ) =
=
+
+
And
=
+
+
Or,
=
+
+
∴
r eq = r
Putting in equation (i),
i =
=
+
+
i =
=
∴ E eq = E
So, the equivalent circuit is,
(i) Current flowing through the circuit= i =
(ii) Terminal potential difference,V = i R
Or, V = (
) R
∴ V =
When electron collides with lattice, its velocity becomes instantaneously zero and then again it starts moving in a specific direction due to the applied electric field.
If the average time between two collisions (relaxation time) is
Where
Thus, free electrons in a metal at constant temperature under the action of an electric field attain a constant average velocity.
(b) Since, the wires are joined in series current flowing through then will be same.
Let the current in both
Diameter being same, there areas of cross section are also same. Let it be
So, in wire
In wire
Taking the ratio,
OR
(b) (i) Terminal voltage vs. load resistance graph:
And
∴
Or,
∴
Terminal voltage vs. current graph:
(ii) Three cells combination diagram is given below,
Current through cells
And
Or,
Putting in equation (i),
So, the equivalent circuit is,
(i) Current flowing through the circuit
(ii) Terminal potential difference,
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