Same material: This implies that the resistivity (
ρ ) and the number density of free electrons (
n ) are the same for both wires. Also, the charge of an electron
(e) is a constant.
Lengths ratio:
L1:L2=2:3, which means
L1∕L2=2∕3.
Radii ratio:
r1:r2=1:2.
Connected in parallel to a battery: This means the potential difference (voltage
V ) across both wires is the same, i.e.,
V1=V2=V.
We need to find the ratio of the drift velocities
(vd1:vd2).
The relationship between current (
I ), number density of free electrons (
n ), cross-sectional area (
A ), charge of an electron (
e ), and drift velocity
(vd) is given by:
I=nAevdFrom this, the drift velocity can be expressed as:
vd= (Equation 1 ) According to Ohm's Law, the current in a wire is related to the potential difference
(V) and resistance
(R) :
I=( Equation 2)The resistance of a wire is given by:
R= (Equation 3) where
ρ is the resistivity,
L is the length, and
A is the cross-sectional area.
Now, substitute Equation 3 into Equation 2:
I== (Equation 4) Finally, substitute Equation 4 into Equation 1 for
I :
vd=Notice that the cross-sectional area
(A) appears in both the numerator and the denominator, so it cancels out:
vd=This derived formula shows that the drift velocity
(vd) depends on the applied voltage
(V), the number density of free electrons
(n), the electron charge
(e), the resistivity
(ρ), and the length of the wire
(L).
Since the wires are made of the same material,
n and
ρ are constant. The charge of an electron
e is also a constant. When connected in parallel, the voltage
V across both wires is the same.
Therefore, for these two wires,
V,n,e and
ρ are all constant. This means the drift velocity is inversely proportional to the length of the wire:
vd∝So, the ratio of the drift velocities will be:
==We are given the ratio of lengths
L1:L2=2:3.
This means
=.
Therefore,
=.
Substituting this into the ratio of drift velocities:
=The ratio of the drift velocities is
3:2.
Note that the information about the radii ratio was not needed for this calculation, as the cross-sectional area cancels out in the final expression for drift velocity when the voltage across the wire is constant.
The final answer is
3:2.