Gausss-Lawin-Electrostatics L-3 Gauss’s Law in Electrostatics
→ \rightarrow → → \rightarrow → Gauss's Law in Electrostatics → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Gauss’s Law
Relates Electric fields & Charges
FLUX : Flow in Latin
Electric flux and charges
→ \rightarrow → → \rightarrow → Gauss's Law → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3
Gauss's Law in Electrostatics → \rightarrow → → \rightarrow → Fluid Flowing with Uniform Velocity → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Volume Rate of Flow
S: Surface area
Volume rate of flow: v.S
Flux: V.S
Gauss's Law → \rightarrow → → \rightarrow → Volume Rate of Flow → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Flux
volume : v S cos θ vS \cos \theta v S cos θ
flux: v S cos θ = v ⃗ ⋅ n ^ S v S \cos \theta=\vec{v} \cdot \hat{n}S v S cos θ = v ⋅ n ^ S
n ^ \hat{n} n ^ S S S
Fluid Flowing with Uniform Velocity → \rightarrow → → \rightarrow → Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Vector Area
Vecter Area: S → = S n ^ \overrightarrow{S}=S \hat{n} S = S n ^
Volume Rate of Flow → \rightarrow → → \rightarrow → Vector Area → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Vector Area
Flux → \rightarrow → → \rightarrow → Vector Area → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Electric Flux
Uniform Electric Field
Electric flux: Φ = E S = E ⃗ ⋅ S ⃗ \begin{aligned} \Phi & =E S \ & =\vec{E} \cdot \vec{S}\end{aligned} Φ = ES = E ⋅ S
Vector Area → \rightarrow → → \rightarrow → Electric Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Electric Field
Vector Area → \rightarrow → → \rightarrow → Electric Field → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Electric Flux Calculation
Electric Flux → \rightarrow → → \rightarrow → Electric Flux Calculation → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Electric Flux Calculation
E ⃗ = E 0 j ^ \vec{E}=E_0 \hat{j} E = E 0 j ^
Flux Φ 1 = E ⃗ ⋅ S ⃗ = E 0 j ^ ⋅ S j ^ \Phi_1=\vec{E} \cdot \vec{S} =E_0 \hat{j} \cdot S \hat{j} Φ 1 = E ⋅ S = E 0 j ^ ⋅ S j ^
(HCDI) = E 0 S \text { (HCDI) } =E_0 S (HCDI) = E 0 S
Flux Φ 2 = E ⃗ ⋅ S ⃗ = E 0 j ^ [ S j ^ ] \text { Flux } \Phi_2=\vec{E} \cdot \vec{S} =E_0 \hat{j}[ S \hat{j}] Flux Φ 2 = E ⋅ S = E 0 j ^ [ S j ^ ]
(AFGB) = − E 0 S \text { (AFGB) } = -E_0 S (AFGB) = − E 0 S
Electric Field → \rightarrow → → \rightarrow → Electric Flux Calculation → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Total Electric Flux
Flux Through(BCHG) = Φ 3 = E ⃗ ⋅ S ⃗ \text { Flux Through(BCHG) }=\Phi_3=\vec{E} \cdot \vec{S} Flux Through(BCHG) = Φ 3 = E ⋅ S
= E 0 j ^ ⋅ S i ^ = 0 =E_0 \hat{j} \cdot S \hat{i} =0 = E 0 j ^ ⋅ S i ^ = 0 Total Electric flux = E 0 S − E 0 S = 0 \text { Total Electric flux} = E_0 S -E_0 S = 0 Total Electric flux = E 0 S − E 0 S = 0
Electric Flux Calculation → \rightarrow → → \rightarrow → Total Electric Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Electric Flux Calculation
The cube is now rotated-
Face AFGB makes an angle θ \theta θ x x x
Electric Flux Calculation → \rightarrow → → \rightarrow → Electric Flux Calculation → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Vector Areas of Surfaces
Total Electric Flux → \rightarrow → → \rightarrow → Vector Areas of Surfaces → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Electric Flux
Electric flux B C H G B C H G BC H G
Φ 1 = E ⃗ ⋅ S ⃗ = E 0 j ^ ⋅ S ( i ^ cos θ + j ^ sin θ ) = E 0 S sin θ \Phi_1 =\vec{E} \cdot \vec{S} =E_0 \hat{j} \cdot S(\hat{i} \cos \theta+\hat{j} \sin \theta) =E_0 S \sin \theta Φ 1 = E ⋅ S = E 0 j ^ ⋅ S ( i ^ cos θ + j ^ sin θ ) = E 0 S sin θ
Flux through(ADIF) \text { Flux through(ADIF) } Flux through(ADIF)
Φ 2 = E ⃗ ⋅ S → {\Phi}_2 =\vec{E} \cdot \overrightarrow{S} Φ 2 = E ⋅ S
= E 0 j ^ ⋅ S ( − i ^ cos θ − j ^ sin θ ) E_0 \hat{j} \cdot S(-\hat{i} \cos {\theta} -\hat{j} \sin {\theta}) E 0 j ^ ⋅ S ( − i ^ cos θ − j ^ sin θ )
= − S E 0 sin θ =-S E_0\sin \theta = − S E 0 sin θ
Electric Flux Calculation → \rightarrow → → \rightarrow → Electric Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Flux
Φ 3 = E ⃗ S ⃗ = E 0 j ^ ⋅ S ( − i ^ sin θ + j ^ cos θ ) = E 0 S cos θ
\Phi_3 =\vec{E} \vec{S} =E_0 \hat{j} \cdot S(-\hat{i} \sin \theta+\hat{j} \cos \theta) =E_0 S \cos \theta Φ 3 = E S = E 0 j ^ ⋅ S ( − i ^ sin θ + j ^ cos θ ) = E 0 S cos θ
Φ 4 = E ⃗ ⋅ S → = E 0 j ^ ⋅ S ( i ^ sin θ − j ^ cos θ ) = − E 0 S cos θ
\Phi_4 =\vec{E} \cdot \overrightarrow{S} =E_0 \hat{j} \cdot S(\hat{i} \sin \theta-\hat{j} \cos \theta) = -E_0 S \cos \theta Φ 4 = E ⋅ S = E 0 j ^ ⋅ S ( i ^ sin θ − j ^ cos θ ) = − E 0 S cos θ
Flux through top and bottom surfaces = 0 = 0 = 0
Vector Areas of Surface → \rightarrow → → \rightarrow → Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Total Flux
Total flux = E 0 S sin θ − E 0 S sin θ + E 0 S cos θ − E 0 S cos θ = 0 \text { Total flux } =E_0 S \sin \theta - E_0 S \sin \theta + E_0 S \cos \theta - E_0 S \cos \theta = 0 Total flux = E 0 S sin θ − E 0 S sin θ + E 0 S cos θ − E 0 S cos θ = 0
Electric Flux → \rightarrow → → \rightarrow → Total Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Electric Flux through Sphere
Sphere of radius R R R
E ⃗ = 1 4 π ϵ 0 q r 2 r ^ \vec{E}=\frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \hat{r} E = 4 π ϵ 0 1 r 2 q r ^
d s ⃗ d \vec{s} d s
d ϕ = E ⃗ ⋅ d s ⃗ d \phi=\vec{E} \cdot d \vec{s} d ϕ = E ⋅ d s
E = 1 4 π ϵ 0 q R 2 E=\frac{1}{4 \pi \epsilon_0} \frac{q}{R^2} E = 4 π ϵ 0 1 R 2 q
Φ = E ⋅ 4 π R 2 \Phi =E \cdot 4 \pi R^2 Φ = E ⋅ 4 π R 2
Total flux = q / ϵ 0 =q / \epsilon_0 = q / ϵ 0
Flux → \rightarrow → → \rightarrow → Electric Flux through Sphere → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Electric Flux through Arbitrary Surface
Electric flux passing through the spherical surface S 1 S_1 S 1 S 2 S_2 S 2
Gauss’s Law :
∮ E ⃗ ⋅ d A ⃗ = Q enc ϵ 0 \oint \vec{E} \cdot d \vec{A}=\frac{Q_{\text {enc }}}{\epsilon_0} ∮ E ⋅ d A = ϵ 0 Q enc
Total Flux → \rightarrow → → \rightarrow → Electric Flux through Arbitrary Surface → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Flux
Φ = q ϵ 0 \Phi=\frac{q}{\epsilon_0} Φ = ϵ 0 q Gauss’s law Φ = q ϵ 0 \text { Gauss’s law } \Phi=\frac{q}{\epsilon_0} Gauss’s law Φ = ϵ 0 q
Electric Flux through Sphere → \rightarrow → → \rightarrow → Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Flux
Φ = q ϵ 0 \Phi=\frac{q}{\epsilon_0} Φ = ϵ 0 q Total Electric flux
Φ = q 1 ϵ 0 + q 2 ϵ 0 \Phi =\frac{q_1}{\epsilon_0}+\frac{q_2}{\epsilon_0} Φ = ϵ 0 q 1 + ϵ 0 q 2
= Σ q 1 ϵ 0 = Q ϵ 0 =\frac {\Sigma q_1}{\epsilon_0} =\frac{Q}{\epsilon_0} = ϵ 0 Σ q 1 = ϵ 0 Q
Total charges enclosed by the surfaces
Electric Flux through Arbitrary Surface → \rightarrow → → \rightarrow → Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Total Electric Flux
Φ ˉ = q 1 + q 2 + q 3 ϵ 0 \bar{\Phi} =\frac{q_1 + q_2+ q_3}{\epsilon_0} Φ ˉ = ϵ 0 q 1 + q 2 + q 3
= ∑ i q i ϵ 0 =\frac{\sum_i q_i}{\epsilon_0} = ϵ 0 ∑ i q i
∑ q L : \sum q_L: ∑ q L :
Total flux = ( q 1 + q 2 ) ϵ 0 \text { Total flux}=\frac{\left(q_1+q_2\right)}{\epsilon_0} Total flux = ϵ 0 ( q 1 + q 2 )
Φ = charg enclosed ϵ 0 \Phi=\frac{\text { charg enclosed }}{\epsilon_0} Φ = ϵ 0 charg enclosed
Flux → \rightarrow → → \rightarrow → Total Electric Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Total Flux
Flux → \rightarrow → → \rightarrow → Total Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Conductor
Total Electric Flux → \rightarrow → → \rightarrow → Conductor → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Flux
Total Flux → \rightarrow → → \rightarrow → Flux → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Conducting Sphere with Charge +Q
Surface charge density σ = Q 4 π R 2 \sigma=\frac{Q}{4 \pi R^2} σ = 4 π R 2 Q
Conductor → \rightarrow → → \rightarrow → Conducting Sphere with Charge +Q → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Problem
Flux → \rightarrow → → \rightarrow → Problem → \rightarrow → → \rightarrow →
Gausss-Lawin-Electrostatics L-3 Thank You
Conducting Sphere with Charge +Q → \rightarrow → → \rightarrow → Thankyou → \rightarrow → → \rightarrow →
Resume presentation
Gausss-Lawin-Electrostatics L-3 Gauss’s Law in Electrostatics $\rightarrow$ $\rightarrow$ Gauss's Law in Electrostatics $\rightarrow$ Gauss's Law $\rightarrow$ Fluid Flowing with Uniform Velocity
