Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Energy Stored in Capacitors Gauss’s Law in Dielectrics
→ \rightarrow → → \rightarrow → Energy Stored in Capacitors Gauss's Law in Dielectrics → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Electrostatics Energy Stored in a Capacitor
→ \rightarrow → → \rightarrow → Electrostatics Energy Stored in a Capacitor → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Work Done in Moving Charge
Work done in moving charge dq = Vdq = q d q c \frac{qdq}{c} c q d q
Total work done W= ∫ 0 Q q c . d q \int_{0}^{Q}\frac{q}{c}.dq ∫ 0 Q c q . d q 1 c ∫ 0 Q q d q \frac{1}{c}\int_{0}^{Q} qdq c 1 ∫ 0 Q q d q Q 2 2 C \frac{Q^2}{2C} 2 C Q 2
Energy Stored = U = Q 2 2 C \frac{Q^2}{2C} 2 C Q 2
Energy Stored in Capacitors Gauss's Law in Dielectrics → \rightarrow → → \rightarrow → Work Done in Moving Charge → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Work Done in Moving Charge
U = Q 2 2 C = 1 2 C C 2 v 2 = 1 2 C v 2 U=\frac{Q^2}{2 C}=\frac{1}{2 C} C^2 v^2=\frac{1}{2} C v^2 U = 2 C Q 2 = 2 C 1 C 2 v 2 = 2 1 C v 2
U = Q 2 C c v = 1 2 Q v U=\frac{Q}{2 C} c v=\frac{1}{2} Q v U = 2 C Q c v = 2 1 Q v
U = Q 2 2 C ; U = 1 2 C v 2 ; U = 1 2 Q v U=\frac{Q^2}{2 C} ; U=\frac{1}{2} C v^2 ; U=\frac{1}{2} Q v U = 2 C Q 2 ; U = 2 1 C v 2 ; U = 2 1 Q v
Electrostatics Energy Stored in a Capacitor → \rightarrow → → \rightarrow → Work Done in Moving Charge → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Parallel Plate Capacitor
U = Q 2 2 C = 1 2 C v 2 U=\frac{Q^2}{2 C}=\frac{1}{2} C v^2 U = 2 C Q 2 = 2 1 C v 2
C = ϵ 0 A d \frac{\epsilon_0 A}{d} d ϵ 0 A
V = E d V=Ed V = E d
U = 1 2 ϵ 0 A d ⋅ E 2 d 2 U=\frac{1}{2} \frac{\epsilon_0 A}{d} \cdot E^2 d^2 U = 2 1 d ϵ 0 A ⋅ E 2 d 2 = ( 1 2 ϵ 0 E 2 ) ⋅ ( A d ) =\left(\frac{1}{2} \epsilon_0 E^2\right) \cdot (Ad) = ( 2 1 ϵ 0 E 2 ) ⋅ ( A d )
(Ad)= Volume Enclosed
Work Done in Moving Charge → \rightarrow → → \rightarrow → Parallel Plate Capacitor → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Example
Spherical Capacitor
C = 4 π ϵ 0 r a r b ( r b − r a ) C=4 \pi \epsilon_0 \frac{r_a r_b}{\left(r_b-r_a\right)} C = 4 π ϵ 0 ( r b − r a ) r a r b
Energy Stored
U = 1 2 Q 2 C = Q 2 ( r b − r a ) 8 π ϵ 0 r a r b U=\frac{1}{2} \frac{Q^2}{C}=\frac{Q^2\left(r_b-r_a\right)}{8 \pi \epsilon_0 r_a r_b} U = 2 1 C Q 2 = 8 π ϵ 0 r a r b Q 2 ( r b − r a )
Work Done in Moving Charge → \rightarrow → → \rightarrow → Example → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Energy Density
Energy Density = 1 2 ϵ 0 E 2 \frac{1}{2}\epsilon_0E^2 2 1 ϵ 0 E 2
4 π r 2 E = Q ϵ 0 4 \pi r^2 E=\frac{Q}{\epsilon_0} 4 π r 2 E = ϵ 0 Q
E = Q 4 π ϵ 0 r 2 E=\frac{Q}{4 \pi \epsilon_0 r^2} E = 4 π ϵ 0 r 2 Q
d U = 1 2 ϵ 0 Q 2 ( 4 π ϵ 0 ) 2 r 4 4 π r 2 d r d U=\frac{1}{2} \epsilon_0 \frac{Q^2}{\left(4 \pi \epsilon_0\right)^2 r^4} 4 \pi r^2 d r d U = 2 1 ϵ 0 ( 4 π ϵ 0 ) 2 r 4 Q 2 4 π r 2 d r = Q 2 8 π ϵ 0 d r r 2 =\frac{Q^2}{8 \pi \epsilon_0} \frac{d r}{r^2} = 8 π ϵ 0 Q 2 r 2 d r
Parallel Plate Capacitor → \rightarrow → → \rightarrow → Energy Density → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Total Stored Energy
U = ∫ r a r b Q 2 8 π ϵ 0 d r r 2 \int_{r_a}^{r_b}\frac{Q^2}{8\pi\epsilon_0}\frac{dr}{r^2} ∫ r a r b 8 π ϵ 0 Q 2 r 2 d r Q 2 8 π ϵ 0 ( 1 r a − 1 r b ) \frac{Q^2}{8\pi\epsilon_0}(\frac{1}{r_a}-\frac{1}{r_b}) 8 π ϵ 0 Q 2 ( r a 1 − r b 1 ) Q 2 ( r b − r a ) 8 π ϵ 0 r a r b \frac{Q^2(r_b-r_a)}{8\pi\epsilon_0r_ar_b} 8 π ϵ 0 r a r b Q 2 ( r b − r a )
Example → \rightarrow → → \rightarrow → Total Stored Energy → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Dielectrics and Polarization
Energy Density → \rightarrow → → \rightarrow → Dielectrics and Polarization → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Bound Surface
Total Stored Energy → \rightarrow → → \rightarrow → Bound Surface → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Polarization
PolarizationP ⃗ \vec{P} P
P ⃗ = ϵ 0 χ E ⃗ \vec{P}={\epsilon_0}\chi \vec{E} P = ϵ 0 χ E
Where, χ \chi χ
Dielectrics and Polarization → \rightarrow → → \rightarrow → Polarization → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Polarization
Polarization = P
Volume = Al
Dipole moment = PAl
Dipole moment = q.l
ql = PAl
q = PA
Bound Surface → \rightarrow → → \rightarrow → Polarization → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Bound Surface Charge Density
σ b = q A = p \sigma_b=\frac{q}{A}=p σ b = A q = p
Area = A c o s θ \frac{A}{cos{\theta}} cos θ A
Bound charge Density
σ b = q A / c o s θ = P c o s θ {\sigma_b}=\frac{q}{A/cos{\theta}}=Pcos{\theta} σ b = A / cos θ q = P cos θ
σ b = P ⃗ ⋅ n ^ \sigma_b=\vec{P} \cdot \hat{n} σ b = P ⋅ n ^
Polarization → \rightarrow → → \rightarrow → Bound Surface Charge Density → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Capacitor with a Dielectric
EA = ( σ f − σ b ) A ϵ 0 \frac{\left(\sigma_f-\sigma_b\right) A}{\epsilon_0} ϵ 0 ( σ f − σ b ) A
ϵ 0 E = σ f − σ b \ \epsilon_0 E=\sigma_f-\sigma_b ϵ 0 E = σ f − σ b
σ b = P = ϵ 0 χ E {\sigma_b}=P={\epsilon_0}{\chi}E σ b = P = ϵ 0 χ E
ϵ 0 E + ϵ 0 χ E = σ f {\epsilon_0}E+{\epsilon_0}{\chi}E={\sigma_f} ϵ 0 E + ϵ 0 χ E = σ f
ϵ 0 ( 1 + χ ) E = σ f {\epsilon_0}(1+{\chi})E={\sigma_f} ϵ 0 ( 1 + χ ) E = σ f
Polarization → \rightarrow → → \rightarrow → Capacitor with a Dielectric → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Dielectric Constant
Dielectric Constant K=1+χ \chi χ
Permittivity of the Dielectric ϵ = ϵ 0 ( 1 + χ ) {\epsilon}={\epsilon_0}(1+{\chi}) ϵ = ϵ 0 ( 1 + χ ) ϵ 0 K {\epsilon_0K} ϵ 0 K
k ⩾ 1 k \geqslant 1 k ⩾ 1
E=σ f ϵ 0 k = σ f ϵ \frac{\sigma_f}{\epsilon_0k}=\frac{\sigma_f}{\epsilon} ϵ 0 k σ f = ϵ σ f
Bound Surface Charge Density → \rightarrow → → \rightarrow → Dielectric Constant → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Dielectric Constant
Material K
Pyrex Glass
4.7
Polystyrene
2.6
Paper
3.5
Porcelain
6.5
Titanium Ceramic
130
Strontium Titante
310
Water
80.4
Capacitor with a Dielectric → \rightarrow → → \rightarrow → Dielectric Constant → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Example
E = σ f ϵ E=\frac{\sigma_f}{\epsilon} E = ϵ σ f
σ f = Q A \sigma_f=\frac{Q}{A} σ f = A Q
E = v α E=\frac{v}{\alpha} E = α v
v α = Q A ϵ \frac{v}{\alpha}=\frac{Q}{A\epsilon} α v = A ϵ Q
Q = c v Q=cv Q = c v
v = Q α A ϵ v=Q\frac{\alpha}{A\epsilon} v = Q A ϵ α
Dielectric Constant → \rightarrow → → \rightarrow → Example → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Capacitance
c = ϵ A α = ϵ 0 K A α c=\frac{\epsilon A}{\alpha}=\frac{\epsilon_0 K A}{\alpha} c = α ϵ A = α ϵ 0 K A
Area = 100 c m 2 =100 \mathrm{~cm}^2 = 100 cm 2
α = 1 c m \alpha= 1{~cm} α = 1 c m
(1) Axis separating the plates ϵ ≃ ϵ 0 \epsilon \simeq \epsilon_0 ϵ ≃ ϵ 0
c a = ϵ A α = 8.85 p F c_a=\frac{\epsilon A}{\alpha}=8.85{~p^F} c a = α ϵ A = 8.85 p F
(2) Dielectric filling K = 2.6 {~K=2.6} K = 2.6
C a = 23.01 p F C_a=23.01{~p^F} C a = 23.01 p F
Dielectric Constant → \rightarrow → → \rightarrow → Capacitance → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Dielectric Filling Capacitance
Example → \rightarrow → → \rightarrow → Delectric Filling Capacitance → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Gauss’s Law in Dielectric
E A = ( σ f − σ b ) A ϵ 0 ({\sigma_f-\sigma_b})\frac{A}{\epsilon_0} ( σ f − σ b ) ϵ 0 A
ϵ 0 E + σ b = σ f \epsilon_0 E+\sigma_b=\sigma_f ϵ 0 E + σ b = σ f
σ b = P \sigma_b=P σ b = P
ϵ 0 E + P = σ f \epsilon_0 E + P = \sigma _f ϵ 0 E + P = σ f
Displacement vector
D ⃗ = ϵ 0 E ⃗ + P ⃗ \vec{D}=\epsilon_0\vec{E}+\vec{P} D = ϵ 0 E + P
Capacitance → \rightarrow → → \rightarrow → Gauss's Law in Dielectric → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Gauss’s Law in Dielectric
D = σ f D = \sigma_f D = σ f
D A = σ f A D A = \sigma_f A D A = σ f A
∮ D ⃗ ⋅ d A ⃗ = Q f enc \oint \vec{D} \cdot d \vec{A}=Q_{f \text { enc }} ∮ D ⋅ d A = Q f enc
∮ ϵ E ⃗ ⋅ d A ⃗ = Q f enc \oint\epsilon \vec{E} \cdot d \vec{A}=Q_{f \text { enc }} ∮ ϵ E ⋅ d A = Q f enc
Delectric Filling Capacitance → \rightarrow → → \rightarrow → Gauss's Law in Dielectric → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Example
∮ D ⃗ ⋅ d A ⃗ = Q f enc \oint \vec{D} \cdot d \vec{A}=Q_{\text {f enc }} ∮ D ⋅ d A = Q f enc
D. 4 π r 2 = Q 4 \pi r^2=Q 4 π r 2 = Q
D = Q 4 π r 2 D=\frac{Q}{4 \pi r^2} D = 4 π r 2 Q
$a
E = Q 4 π ϵ r 2 E=\frac{Q}{4 \pi \epsilon r^2} E = 4 π ϵ r 2 Q
Gauss's Law in Dielectric → \rightarrow → → \rightarrow → Example → \rightarrow → → \rightarrow →
Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Thank You
Gauss's Law in Dielectric → \rightarrow → → \rightarrow → Thankyou → \rightarrow → → \rightarrow →
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Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 Energy Stored in Capacitors Gauss’s Law in Dielectrics $\rightarrow$ $\rightarrow$ Energy Stored in Capacitors Gauss's Law in Dielectrics $\rightarrow$ Electrostatics Energy Stored in a Capacitor $\rightarrow$ Work Done in Moving Charge
