Electrostatic Potential and Potential Energy L-5 Electrostatic Potential and Potential Energy
→ \rightarrow → → \rightarrow → Electrostatic Potential and Potential Energy → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Electrostatic Potential Energy
E ⃗ = − E 0 ı ^ \vec{E}=-E_0 \hat{\imath} E = − E 0 ^
F ⃗ e l = − q E 0 c ^ \vec{F}_{el}=-q E_0 \hat{c} F e l = − q E 0 c ^
F ⃗ e x t = − F ⃗ e l = q E 0 i ^ \vec F_{ext} = -\vec F_{el} = q E_0 \hat{i} F e x t = − F e l = q E 0 i ^
l ⃗ = ( x f − x i ) i ^ \vec l = (x_{f}-x_{i}) \hat i l = ( x f − x i ) i ^
Work done by the external force on the charge = F ⃗ e x t ⋅ l ⃗ =\vec{F}_{ext } \cdot \vec{l} = F e x t ⋅ l
= q E 0 ( x f − x i ) =q E_0\left(x_{f}-x_i\right) = q E 0 ( x f − x i ) = q E 0 h =q E_0 h = q E 0 h
→ \rightarrow → → \rightarrow → Electrostatic Potential Energy → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Electrostatic Potential Energy
E ⃗ = − E 0 ı ^ \vec{E}=-E_0 \hat{\imath} E = − E 0 ^
F ⃗ e l = − q E 0 c ^ \vec{F}_{el}=-q E_0 \hat{c} F e l = − q E 0 c ^
F ⃗ e x t = − F ⃗ e l = q E 0 i ^ \vec F_{ext} = -\vec F_{el} = q E_0 \hat{i} F e x t = − F e l = q E 0 i ^
l ⃗ = ( x f − x i ) i ^ \vec l = (x_{f}-x_{i}) \hat i l = ( x f − x i ) i ^
Work done by the external force on the charge = F ⃗ e x t ⋅ l ⃗ =\vec{F}_{ext } \cdot \vec{l} = F e x t ⋅ l
= q E 0 ( x f − x i ) =q E_0\left(x_{f}-x_i\right) = q E 0 ( x f − x i ) = q E 0 h =q E_0 h = q E 0 h
Electrostatic Potential and Potential Energy → \rightarrow → → \rightarrow → Electrostatic Potential Energy → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Energy
Electrostatic Potential Energy → \rightarrow → → \rightarrow → Energy → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Energy
U f U_f U f U i U_i U i U f U_f U f U i U_i U i
Similar figure for E ⃗ \vec E E
Electrostatic Potential Energy → \rightarrow → → \rightarrow → Energy → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Work Done
Energy → \rightarrow → → \rightarrow → Work Done → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Work Done
Work done in moving the charge from A to C
= F ⃗ e x t l ⃗ \vec F_{ext} \vec{l} F e x t l F e l ⋅ l ⃗ F_{el} \cdot \vec {l} F e l ⋅ l
= +q E 0 ı ^ ⋅ ( z f − z i ) k ^ E_0 \hat{\imath} \cdot \left(z_f-z_i\right) \hat{k} E 0 ^ ⋅ ( z f − z i ) k ^
= 0 c + B c+B \quad c + B = q E 0 ( x f − x i ) = q E_0\left(x_f-x_i\right) = q E 0 ( x f − x i )
Energy → \rightarrow → → \rightarrow → Work Done → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Total Work Done
Total work done = q E 0 ( x f − x i ) E_0 (x_f - x_i) E 0 ( x f − x i )
Path 2 - A → \rightarrow → E 0 ( x f − x i ) E_0 (x_f - x_i) E 0 ( x f − x i )
D → \rightarrow →
Total work done by the external force in moving from A to B via D = q E 0 ( x f − x i ) E_0 (x_f - x_i) E 0 ( x f − x i )
Work Done → \rightarrow → → \rightarrow → Total Work Done → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Topic
Path 3 -l ⃗ = ( x f − x i ) ı ^ + ( z f − z i ) k ^ \vec{l}=\left(x_f-x_i\right) \hat{\imath}+\left(z_f-z_i\right) \hat{k} l = ( x f − x i ) ^ + ( z f − z i ) k ^
F ⃗ e x t \vec F_{ext} F e x t − F ⃗ e l = q E 0 i ^ -\vec F_{e l} = q E_0 \hat{i} − F e l = q E 0 i ^
Work done = F e x t → ⋅ l ⃗ \overrightarrow{F_{ext}} \cdot \vec{l} F e x t ⋅ l
= q E 0 ı ^ ⋅ [ ( x f − x i ) ı ^ + ( z f − z i ) k ^ ] =q E_0 \hat{\imath} \cdot\left[\left(x_f-x_i\right) \hat{\imath}+\left(z_f-z_i\right) \hat{k}\right] = q E 0 ^ ⋅ [ ( x f − x i ) ^ + ( z f − z i ) k ^ ] = q E 0 ( x f − x i ) =q E_0\left(x_f-x_i\right) = q E 0 ( x f − x i )
Work done is independent of pair followed
Work Done → \rightarrow → → \rightarrow → Topic → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Example
Point charge Q
F ⃗ el = q E ⃗ \vec{F}_{\text {el}} = q \vec{E} F el = q E = − q Q 4 π E 0 r 2 r ^ =-\frac{q Q}{4 \pi E_0 r^2} \hat{r} = − 4 π E 0 r 2 qQ r ^
f ⃗ e x t = f ⃗ e l \vec f_{ext} = \vec f_{el} f e x t = f e l = − q Q 4 π E 0 r 2 r ^ =-\frac{q Q}{4 \pi E_0 r^2} \hat{r} = − 4 π E 0 r 2 qQ r ^
Work done = ∫ r i r f F ⃗ . d l ⃗ \int^{r_f}_{r_i} \vec F .\vec {dl} ∫ r i r f F . d l
d l ⃗ = d r ⃗ \vec {dl} = d \vec{r} d l = d r r ^ . d r \hat r .d r r ^ . d r
Total Work Done → \rightarrow → → \rightarrow → Example → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Work Done by External Force
Work done by external force = ∫ r i r f F e x t ⋅ d l → =\int_{r_i}^{r_f} F_{ext} \cdot \overrightarrow{dl} = ∫ r i r f F e x t ⋅ d l
= ∫ r i r f ( − q Q 4 π E 0 r 2 r ^ ) ⋅ d r r ^ =\int_{r_i}^{r_f}\left(-\frac{q Q}{4 \pi E_0 r^2} \hat{r}\right) \cdot d r \hat{r} = ∫ r i r f ( − 4 π E 0 r 2 qQ r ^ ) ⋅ d r r ^
= − q Q 4 π E 0 ∫ r i r f d r r 2 =-\frac{q Q}{4 \pi E_0} \int_{r_i}^{r_f} \frac{d r}{r^2} = − 4 π E 0 qQ ∫ r i r f r 2 d r
= − q Q 4 π E 0 ( − 1 r ) ∣ r i r f =-\left.\frac{q Q}{4 \pi E_0}\left(-\frac{1}{r}\right)\right|_{r_i} ^{r_f} = − 4 π E 0 qQ ( − r 1 ) r i r f
= q Q 4 π E 0 ( 1 r f − 1 r i ) =\frac{q Q}{4 \pi E_0}\left(\frac{1}{r_f}-\frac{1}{r_i}\right) = 4 π E 0 qQ ( r f 1 − r i 1 )
Topic → \rightarrow → → \rightarrow → Work Done by External Force → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Work Done
U B < U A U_B < U_A U B < U A
Work done = q Q 4 π E 0 ( 1 r f − 1 r i ) =\frac{q Q}{4 \pi E_0}\left(\frac{1}{r_f}-\frac{1}{r_i}\right) = 4 π E 0 qQ ( r f 1 − r i 1 )
q & Q both +ve
( r f > r i ) (r_f > r_i) ( r f > r i )
Work done < 0 <0 < 0
Work done = ( − q ) Q 4 π E 0 ( 1 r f − 1 r i ) \frac{(-q) Q}{4 \pi E_0}\left(\frac{1}{r_f}-\frac{1}{r_i}\right) 4 π E 0 ( − q ) Q ( r f 1 − r i 1 )
U B > U A U_B > U_A U B > U A
Example → \rightarrow → → \rightarrow → Work Done → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Work Done
Work Done by External Force → \rightarrow → → \rightarrow → Work Done → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Work Done
∫ F ⃗ e x t ⋅ d l → \int \vec{F}_{ext} \cdot \overrightarrow{d l} ∫ F e x t ⋅ d l
=∫ A C 1 B F ⃗ e x t ⋅ d l → \int_{A_{C_1}}^{B} \vec{F}_{ext} \cdot \overrightarrow{d l} ∫ A C 1 B F e x t ⋅ d l
+ ∫ B c 2 A F ⃗ e x t ⋅ d l → +\int_{B_{c_2}}^{A} \vec{F}_{ext} \cdot \overrightarrow{d l} + ∫ B c 2 A F e x t ⋅ d l
∫ A C 1 B F ⃗ e x t ⋅ d l → \int_{A_{C_1}}^{B} \vec{F}_{ext} \cdot \overrightarrow{d l} ∫ A C 1 B F e x t ⋅ d l
=∫ A C 2 B F ⃗ e x t ⋅ d l → \int_{A_{C_2}}^{B} \vec{F}_{ext} \cdot \overrightarrow{d l} ∫ A C 2 B F e x t ⋅ d l
=− ∫ B c 2 A F ⃗ e x t ⋅ d l → -\int_{B_{c_2}}^{A} \vec{F}_{ext} \cdot \overrightarrow{d l} − ∫ B c 2 A F e x t ⋅ d l
Work Done → \rightarrow → → \rightarrow → Work Done → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Work Done
∮ F ⃗ e x t ⋅ d l → \oint \vec{F}_{ext} \cdot \overrightarrow{d l} ∮ F e x t ⋅ d l
Not work done in a closed path = 0
Work Done → \rightarrow → → \rightarrow → Work Done → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Electrostatic Potential Energy
Work Done → \rightarrow → → \rightarrow → Electrostatic Potential Energy → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Potential Energy
Potential energy U = Q q 4 π E 0 r \frac {Q q} {4 \pi E_0 r} 4 π E 0 r Qq
Work Done → \rightarrow → → \rightarrow → Potential Energy → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Potential Energy
U 12 = Q q 4 π E 0 r U_{12} = \frac {Q q} {4 \pi E_0 r} U 12 = 4 π E 0 r Qq
W = ∫ ∞ C F ⃗ e x t ⋅ d l → \int_{\infty}^{C} \vec{F}_{ext} \cdot \overrightarrow{d l} ∫ ∞ C F e x t ⋅ d l
= - ∫ ∞ C F ⃗ e l ⋅ d l → \int_{\infty}^{C} \vec{F}_{el} \cdot \overrightarrow{d l} ∫ ∞ C F e l ⋅ d l
= F ⃗ e l = q 3 E ⃗ 1 + q 3 E ⃗ 2 \vec F_{el} = q_3 \vec E_1 + q_3 \vec E_2 F e l = q 3 E 1 + q 3 E 2
Electrostatic Potential Energy → \rightarrow → → \rightarrow → Potential Energy → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Total Potential Energy
W =− q 3 ∫ ∞ C E ⃗ 1 ⋅ d l → -q_ 3\int_ {\infty}^{C} \vec{E}_ {1} \cdot \overrightarrow{d l} − q 3 ∫ ∞ C E 1 ⋅ d l q 3 ∫ ∞ C E ⃗ 2 ⋅ d l → q_ 3\int_ {\infty}^{C} \vec{E}_ {2} \cdot \overrightarrow{d l} q 3 ∫ ∞ C E 2 ⋅ d l
= q 1 q 3 4 π E 0 r 13 + q 2 q 3 4 π E 0 r 23 \frac {q_1 q_3} {4 \pi E_{0} r_{13}} + \frac {q_2 q_3} {4 \pi E_{0} r_{23}} 4 π E 0 r 13 q 1 q 3 + 4 π E 0 r 23 q 2 q 3
Total potential energy, U = q 1 q 2 4 π E 0 r 12 + q 1 q 3 4 π E 0 r 13 + q 2 q 3 4 π E 0 r 23 \frac {q_1 q_2} {4 \pi E_{0} r_{12}} + \frac {q_1 q_3} {4 \pi E_{0} r_{13}} + \frac {q_2 q_3} {4 \pi E_{0} r_{23}} 4 π E 0 r 12 q 1 q 2 + 4 π E 0 r 13 q 1 q 3 + 4 π E 0 r 23 q 2 q 3
Electrostatic Potential Energy → \rightarrow → → \rightarrow → Total Potential Energy → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Electrostatic Potential
Work done by an external force in bringing a unit potential charge from uniformily to the point in the electrostatic potential at that point.
U = q Q 4 π E 0 ( 1 r f − 1 r i ) \frac{q Q}{4 \pi E_0}\left(\frac{1}{r_f}-\frac{1}{r_i}\right) 4 π E 0 qQ ( r f 1 − r i 1 ) Q q 4 π E 0 r \frac {Q q} {4 \pi E_0 r} 4 π E 0 r Qq
r i = ∞ r_i = \infty r i = ∞
Potential Energy → \rightarrow → → \rightarrow → Electrostatic Potential → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Point Charge
V (r)= Q 4 π E 0 r \frac {Q} {4 \pi E_0 r} 4 π E 0 r Q Point charge
Total Potential Energy → \rightarrow → → \rightarrow → Point Charge → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Work Done
Work done by the external force to a unit +ve charge from r i r_i r i r f r_f r f V ( r f ) − V ( r i ) V(r_f) - V(r_i) V ( r f ) − V ( r i )
Electrostatic Potential → \rightarrow → → \rightarrow → Work Done → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Unit Potential
(Alessandro Volta 1745-1827)
Electric field : V m \frac {V}{m} m V
Point Charge → \rightarrow → → \rightarrow → Unit Potential → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Unit Potential
(Alessandro Volta 1745-1827)
Electric field : V m \frac {V}{m} m V
Work Done → \rightarrow → → \rightarrow → Unit Potential → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Unit Potential
Unit Potential → \rightarrow → → \rightarrow → Unit Potential → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Unit Potential
Potential: V(A) = q 1 4 π E 0 r + q 2 4 π E 0 r = 0 \frac {q_1}{4\pi E_0 r} + \frac{q_2}{4\pi E_0 r} = 0 4 π E 0 r q 1 + 4 π E 0 r q 2 = 0
V(B) = q 1 4 π E 0 r 1 − q 2 4 π E 0 r 2 \frac {q_1}{4\pi E_0 r_1} - \frac{q_2}{4\pi E_0 r_2} 4 π E 0 r 1 q 1 − 4 π E 0 r 2 q 2
q 2 = − q 1 q_2 = - q_1 q 2 = − q 1
Unit Potential → \rightarrow → → \rightarrow → Unit Potential → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Problem
V ( B ) = 10 × 10 − 9 × 9 × 10 9 4 × 10 − 2 − 10 × 10 − 9 × 9 × 10 9 10 × 10 − 2 V(B) =\frac{10 \times 10^{-9} \times 9 \times 10^9}{4 \times 10^{-2}}-\frac{10 \times 10^{-9} \times 9 \times 10^9}{10 \times 10^{-2}} V ( B ) = 4 × 1 0 − 2 10 × 1 0 − 9 × 9 × 1 0 9 − 10 × 1 0 − 2 10 × 1 0 − 9 × 9 × 1 0 9
= 2.25 × 10 3 − 9 × 10 2 = 1.35 × 10 3 V =2.25 \times 10^3-9 \times 10^2 =1.35 \times 10^3 \mathrm{~V} = 2.25 × 1 0 3 − 9 × 1 0 2 = 1.35 × 1 0 3 V
V(C) = − 1.25 × 10 3 V =-1.25 \times 10^3 \mathrm{V} = − 1.25 × 1 0 3 V
Problem: Calculate the work done in moving a charge of 5 nc from A to B and from A to C
Unit Potential → \rightarrow → → \rightarrow → Problem → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy L-5 Potential of a Charges
Potential of a charges conducting sphere.
Unit Potential → \rightarrow → → \rightarrow → Potential of a Charges → \rightarrow → → \rightarrow →
Electrostatic Potential and Potential Energy
L-5 Thank You
Problem → \rightarrow → → \rightarrow → Thankyou → \rightarrow → → \rightarrow →
Resume presentation
Electrostatic Potential and Potential Energy L-5 Electrostatic Potential and Potential Energy $\rightarrow$ $\rightarrow$ Electrostatic Potential and Potential Energy $\rightarrow$ Electrostatic Potential Energy $\rightarrow$ Electrostatic Potential Energy
