Drift Velocity And Resistance L-2 Drift Velocity and Resistance
→ \rightarrow → → \rightarrow → Drift Velocity and Resistance → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Properties of Charge
1. Current is flow of electric charge.
2. Ability to conduct - depends on material properties - Conductors.
3. "Free electrons".
4. Electric field inside ( under dynamic condition) ≠ 0.
→ \rightarrow → → \rightarrow → Properties of Charge → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Current Density and Drift Velocity
5. Current density J ⃗ \vec J J
I = ∫ J ⃗ ⋅ d s → I=\int \vec{J} \cdot \overrightarrow{d s} I = ∫ J ⋅ d s
6. Electrons
7. Direction of current - direction of flow of positive charge.
8. "Drift velocity"
J ⃗ = − e n v ⃗ α \vec J = -en \vec v_\alpha J = − e n v α
Drift Velocity and Resistance → \rightarrow → → \rightarrow → Current Density and Drift Velocity → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Example
Properties of Charge → \rightarrow → → \rightarrow → Example → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Example
J= I A r e a = 1.5 10 − 7 \frac{I}{Area} = \frac{1.5}{10^{-7}} A re a I = 1 0 − 7 1.5
= 1.5 × 10 7 A / m 2 = 1.5 \times 10^{7} A/m^{2} = 1.5 × 1 0 7 A / m 2
J=− n e v α ⃗ -n e \vec{v_\alpha} − n e v α
1 mol of Cu has a mass 63.5 × 10 − 3 k g 63.5 \times 10^{-3} kg 63.5 × 1 0 − 3 k g
has N = 6 × 10 23 N=6 \times 10^{23} N = 6 × 1 0 23
Current Density and Drift Velocity → \rightarrow → → \rightarrow → Example → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Example
No. of mols in 1 m 3 1 m^{3} 1 m 3
=9 × 10 3 63.5 × 10 − 3 \frac {9 \times 10^3}{63.5 \times 10^{-3}} 63.5 × 1 0 − 3 9 × 1 0 3
No. of atoms =9 × 10 3 63.5 × 10 − 3 × 6.2 × 10 23 \frac{9 \times 10^3}{63.5 \times 10^{-3}} \times 6.2 \times 10^{23} 63.5 × 1 0 − 3 9 × 1 0 3 × 6.2 × 1 0 23
≈ 8.5 × 10 28 m 3 ≡ n \approx 8.5 \times 10^{28} m^{3} \equiv n ≈ 8.5 × 1 0 28 m 3 ≡ n
v α = J n e v_{\alpha} = \frac{J}{n e} v α = n e J
= 1.5 × 10 7 8.5 × 10 28 × 1.6 × 10 − 19 = \frac{1.5 \times 10^{7}}{8.5 \times 10^{28} \times 1.6 \times 10^{-19}} = 8.5 × 1 0 28 × 1.6 × 1 0 − 19 1.5 × 1 0 7
= 1.1 × 10 − 3 m / s = 1.1 m m / s 1.1 \times 10^{-3} m/s = 1.1 {mm} /{s} 1.1 × 1 0 − 3 m / s = 1.1 mm / s
Example → \rightarrow → → \rightarrow → Example → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Boltzmann constant
Average speed of electrons ∼ 10 6 m / s \sim 10^6 m/s ∼ 1 0 6 m / s
Thermal speed of Cu atoms.
1 2 m v t h 2 = 3 2 k B T \frac{1}{2} m v_{t h}^2= \frac{3}{2} k_B T 2 1 m v t h 2 = 2 3 k B T
v t h ≈ K B t m v_{t h} \approx \sqrt{\frac{K_Bt}{m}} v t h ≈ m K B t
K B K_B K B
= 1.38 × 10 − 23. m 2 k g s 2 K = 1.38 \times 10^-23. \frac {m^{2} kg}{s^2 K} = 1.38 × 1 0 − 23. s 2 K m 2 k g
= 1.38 × 10 − 27 × 300 63.5 × 10 − 3 / 6.2 × 10 23 =\sqrt{\frac{1.38 \times 10^{-27} \times 300} { 63.5 \times 10^{-3}/ 6.2\times 10^{23}}} = 63.5 × 1 0 − 3 /6.2 × 1 0 23 1.38 × 1 0 − 27 × 300
≈ 2 × 10 − 2 m / s \approx 2 \times 10^{-2} m/s ≈ 2 × 1 0 − 2 m / s
V d < < V t h V_d << V_{th} V d << V t h
Example → \rightarrow → → \rightarrow → Boltzmann constant → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Ohm's Law
Electric field ∼ 3 × 10 8 m / s \sim 3 \times 10^8 m /s ∼ 3 × 1 0 8 m / s
Drift speed is small
V d ∝ E V_d \propto E V d ∝ E
J ∝ V d J \propto V_d J ∝ V d
J ∝ E J \propto E J ∝ E
J ⃗ = σ E ⃗ , \vec{J} = \sigma \vec {E}, J = σ E , σ \sigma σ
Property of material
S = siemens
J E = A / m 2 V / m = A / v / m \frac {J}{E} = \frac{A/m^2}{V/m} = {A/v/m} E J = V / m A / m 2 = A / v / m
Example → \rightarrow → → \rightarrow → Ohm's Law → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Ohm's Law
J ⃗ {\vec J} J σ E ⃗ {\sigma \vec E} σ E
E ⃗ {\vec E} E ρ J ⃗ {\rho \vec J} ρ J
ρ = 1 σ → o h m − m \rho = \frac {1}{\sigma} \rightarrow ohm-m ρ = σ 1 → o hm − m
1 ohm = 1 volt/A
= S − 1 S^{-1} S − 1
Boltzmann constant → \rightarrow → → \rightarrow → Ohm's Law → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Resistivity of Materials
independent of E ⃗ {\vec E} E
Conductors
Resistivity Ω − m {\Omega -m} Ω − m
Ag
1.6 × 10 − 8 1.6 \times 10^{-8} 1.6 × 1 0 − 8
Cu
1.7 × 10 − 8 1.7 \times 10^{-8} 1.7 × 1 0 − 8
Al
2.75 × 10 − 8 2.75 \times 10^{-8} 2.75 × 1 0 − 8
Insulator
Resistivity Ω − m {\Omega -m} Ω − m
Water
2.5 × 10 5 2.5\times 10^{5} 2.5 × 1 0 5
Glass
10 10 t o 10 14 10^{10} \hspace{2mm} to \hspace{2mm} 10^{14} 1 0 10 t o 1 0 14
Ohm's Law → \rightarrow → → \rightarrow → Resistivity of Materials → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Resistivity of Materials
Resistivity at 0 ∘ C 0^{\circ}C 0 ∘ C
Metal
Resistivity
C(graphite)
∼ 10 − 5 Ω m \sim 10^{-5} \Omega m ∼ 1 0 − 5 Ω m
Ge
: 0.46 Ω m \Omega m Ω m
Si
: 2300Ω m \Omega m Ω m
Ohm's Law → \rightarrow → → \rightarrow → Resistivity of Materials → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Resistance Characterstics
ρ = E J = Δ V / L I / A \rho=\frac{E}{J} =\frac{\Delta V / L}{I / A} ρ = J E = I / A Δ V / L
= ( Δ V I ) × A L . =\left(\frac{\Delta V}{I}\right) \times \frac{A}{L} . = ( I Δ V ) × L A .
Resistance R = ρ × L A R = \rho \times \frac{L}{A} R = ρ × A L
characterstics of a sample
R = Δ V I R=\frac{\Delta V}{I} R = I Δ V
Resistivity of Materials → \rightarrow → → \rightarrow → Resistance Characterstics → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Resistance of Copper
Block of C u \mathrm{Cu} Cu
1 c m × 1 c m × 20 c m 1 cm \times 1cm \times 20cm 1 c m × 1 c m × 20 c m
R = σ L A = 1.3 × 10 − 8 × 20 × 10 − 2 10 − 4 R = \frac {\sigma L}{A} = \frac {1.3 \times 10^{-8} \times 20 \times 10^{-2}}{10^{-4}} R = A σ L = 1 0 − 4 1.3 × 1 0 − 8 × 20 × 1 0 − 2
= 2.6 × 10 − 5 Ω =2.6 \times 10^{-5} \Omega = 2.6 × 1 0 − 5 Ω
Resistance between rectangular ends.
R ′ = 1.3 × 10 − 8 × 10 − 2 20 × 10 − 4 = 0.65 × 10 − 7 Ω R^{\prime}=\frac{1.3 \times 10^{-8} \times 10^{-2}}{20 \times 10^{-4}}=0.65 \times 10^{-7} \Omega R ′ = 20 × 1 0 − 4 1.3 × 1 0 − 8 × 1 0 − 2 = 0.65 × 1 0 − 7 Ω
Resistivity of Materials → \rightarrow → → \rightarrow → Resistance of Copper → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Charge Flow
I = Δ V R = Δ V ρ ⋅ Δ x / A I =\frac{\Delta V}{R}=\frac{\Delta V}{\rho \cdot \Delta x / A} I = R Δ V = ρ ⋅ Δ x / A Δ V
= σ A Δ V Δ x =\sigma A \frac{\Delta V}{\Delta x} = σ A Δ x Δ V
d Q d t = − σ A d V d x \frac{d Q}{d t}=-\sigma A \frac{d V}{d x} d t d Q = − σ A d x d V
Positive charges move in the x direction of decreasing potential.
Resistance Characterstics → \rightarrow → → \rightarrow → Charge Flow → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Charge Flow vs Heat Flow
d Q d t = − x A d T d x \frac{d Q}{d t} = -x A \frac{d T}{d x} d t d Q = − x A d x d T
d T d x \frac{d T}{d x} d x d T
d Q d t \frac{d Q}{d t} d t d Q
x x x
A good conductor of electricity is also a good conductor of heat.
Resistance of Copper → \rightarrow → → \rightarrow → Charge Flow vs Heat Flow → \rightarrow → → \rightarrow → E ⃗ \vec{E} E
Drift Velocity And Resistance L-2 Ohm's Law
V d ∼ a V_d \sim a V d ∼ a m m / s \mathrm{mm} / \mathrm{s} mm / s
ȷ ⃗ = − n e v ⃗ d \vec{\jmath}=-n_e \vec{v}_d = − n e v d
n ∼ 10 28 / m 3 n \sim 10^{28}/ m^{3} n ∼ 1 0 28 / m 3
Why is Ohm's Law good
Electrons collide with ion and emerge with velocity in random direction.
1 N ∑ i v ⃗ i = 0 \frac{1}{N} \sum_i \vec{v}_i=0 N 1 ∑ i v i = 0
Charge Flow → \rightarrow → → \rightarrow → Ohm's Law → \rightarrow → E ⃗ \vec{E} E → \rightarrow →
Drift Velocity And Resistance L-2 In the presence of E ⃗ \vec{E} E
e − e^{-} e − ⟶ \longrightarrow ⟶
→ \rightarrow →
→ \rightarrow →
10 6 m / s ≫ 10 − 3 m / s 10^6 \mathrm{m} / \mathrm{s} \gg 10^{-3} \mathrm{m} / \mathrm{s} 1 0 6 m / s ≫ 1 0 − 3 m / s
Charge Flow vs Heat Flow → \rightarrow → → \rightarrow → In the presence of E ⃗ \vec{E} E → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Collision
a = e E m a=\frac{e E}{m} a = m e E
Time between two successive collision = τ =\tau \quad = τ
Relaxation time
V ⃗ i = \vec{V}_i= V i =
V ⃗ i = v ⃗ i − e E ⃗ m t \vec{V}_i =\vec{v}_i-\frac{e \vec{E}}{m} t V i = v i − m e E t
= v ⃗ i − e E ⃗ τ m =\vec{v}_i-\frac{e \vec{E} \tau}{m} = v i − m e E τ
⟨ v ⃗ i ⟩ = − e E ⃗ ( τ ) m \left\langle\vec{v}_i\right\rangle=-\frac{e \vec{E}(\tau)}{m} ⟨ v i ⟩ = − m e E ( τ )
Ohm's Law → \rightarrow → E ⃗ \vec{E} E → \rightarrow → Collision → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Ohm's Law
J ⃗ = − n e v ⃗ d \vec{J} =-n e \vec{v}_d J = − n e v d
= ( n e 2 τ m ) E ⃗ =\left(\frac{n e^2 \tau}{m}\right) \vec{E} = ( m n e 2 τ ) E
≡ σ E ⃗ \equiv \sigma \vec{E} ≡ σ E
σ = n e 2 τ m \sigma=\frac{n e^2 \tau}{m} σ = m n e 2 τ
→ σ \rightarrow \sigma → σ
→ τ \rightarrow \tau → τ E ⃗ \vec{E} E
In the presence of E ⃗ \vec{E} E → \rightarrow → → \rightarrow → Ohm's Law → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Example
v d ∼ 1.1 × 10 − 3 m / s v_{\mathrm{d}} \sim 1.1 \times 10^{-3} \mathrm{~m} / \mathrm{s} v d ∼ 1.1 × 1 0 − 3 m / s
= e E m . τ = \frac {eE}{m} .\tau = m e E . τ
σ = n e 2 τ m \sigma=\frac{n e^2 \tau}{m} σ = m n e 2 τ
ρ = m n e 2 τ \rho =\frac{m} {ne^{2} \tau} ρ = n e 2 τ m
1.7 × 10 − 8 = 9 × 10 − 31 8.5 × 10 28 × 2.56 × 10 − 38 τ 1.7 \times 10^{-8} =\frac{9 \times 10^{-31}}{8.5 \times 10^{28} \times 2.56 \times 10^{-38} \mathrm{\tau}} 1.7 × 1 0 − 8 = 8.5 × 1 0 28 × 2.56 × 1 0 − 38 τ 9 × 1 0 − 31
τ ≈ 2.4 × 10 − 14 s \tau \approx 2.4 \times 10^{-14} \mathrm{~s} τ ≈ 2.4 × 1 0 − 14 s
Mean Free path
λ = 2.4 × 10 − 14 × 1.6 × 10 6 \lambda =2.4 \times 10^{-14} \times 1.6 \times 10^6 λ = 2.4 × 1 0 − 14 × 1.6 × 1 0 6
≈ 40 n m \approx 40 \mathrm{~nm} ≈ 40 nm
Collision → \rightarrow → → \rightarrow → Example → \rightarrow → → \rightarrow →
Drift Velocity And Resistance L-2 Thank You
Ohm's Law → \rightarrow → → \rightarrow → Thank You → \rightarrow → → \rightarrow →
Resume presentation
Drift Velocity And Resistance L-2 Drift Velocity and Resistance $\rightarrow$ $\rightarrow$ Drift Velocity and Resistance $\rightarrow$ Properties of Charge $\rightarrow$ Current Density and Drift Velocity
