Dielectrics Experiment: Place dielectrics between plates of capacitor at Q=const condition Observation: potential difference decreases to smaller value with dielectric material relative to air 𝐶 0 = 𝑄 𝑉 0 Without dielectric: because V<V0 C>C0 𝑄=𝑐𝑜𝑛𝑠𝑡 𝐶= 𝑄 𝑉 With dielectric: Κ:= 𝐶 𝐶 0 = 𝑉 0 𝑉 K>1: relative dielectric constant

What happens with the E-field in the presence of dielectric material 𝑸=𝒄𝒐𝒏𝒔𝒕 Κ= 𝑉 0 𝑉 = 𝐸 0 𝐸 𝐸= 𝐸 0 Κ We know V<V0 E<E0 specifically d Recall: 𝐸= 𝜎 𝑛𝑒𝑡 𝜖 0 𝜎 𝑛𝑒𝑡 reduced with dielectric material The surface charge (density) σ on conducting plates does not change but induced charge σi of opposite sign 𝐸 0 = 𝜎 𝜖 0 𝐸 = 𝜎− 𝜎 𝑖 𝜖 0 and 𝜎 𝑖 =𝜎(1− 1 𝐾 ) 𝐸= 𝜎 𝐾𝜖 0 Definition of the permittivity 𝜖=𝐾 𝜖 0 𝐶=𝜖 𝐴 𝑑 and 𝑢= 1 2 𝜖 𝐸 2

Dielectrics +Q Example: K1 K2 d/2 E1 E0 V E2 -Q 𝐸 0 = 𝜎 𝜖 0 = 𝑄 𝜖 0 𝐴 𝐸 0 = 𝜎 𝜖 0 = 𝑄 𝜖 0 𝐴 V= 𝐸 1 𝑑 2 + 𝐸 2 𝑑 2 = 𝑄 𝜖 0 𝐴 𝐾 1 + 𝑄 𝜖 0 𝐴 𝐾 2 𝑑 2 = 𝑄𝑑 2 𝜖 0 𝐴 ( 1 𝐾 1 + 1 𝐾 2 ) 𝐸 1 = 𝐸 0 𝐾 1 = 𝑄 𝜖 0 𝐴 𝐾 1 𝐸 2 = 𝐸 0 𝐾 2 = 𝑄 𝜖 0 𝐴 𝐾 2 𝐶= 𝑄 𝑉 = 𝑄 𝑄𝑑 2 𝜖 0 𝐴 ( 1 𝐾 1 + 1 𝐾 2 ) = 2 𝜖 0 𝐴 𝐾 1 𝐾 2 𝑑( 𝐾 1 + 𝐾 2 ) 𝜎 1 =𝜎(1− 1 𝐾 1 ) 𝜎 2 =𝜎(1− 1 𝐾 2 ) 𝑈= 1 2 𝑄𝑉= 𝑄 2 𝑑 4 𝜖 0 𝐴 1 𝐾 1 + 1 𝐾 2 𝑎𝑛𝑑 𝑢= 𝑈 𝐴𝑑 = 1 4 𝜖 0 ( 𝑄 𝐴 ) 2 1 𝐾 1 + 1 𝐾 2 = 𝜖 0 4 ( 𝐸 1 𝐾 1 ) 2 1 𝐾 1 + 1 𝐾 2 = 𝜖 0 4 ( 𝐸 2 𝐾 2 ) 2 1 𝐾 1 + 1 𝐾 2

24.4 Dielectrics High Voltage Cr2O3 Ground High Voltage Air Dielectric breakdown or Dielectric strength

Gauss’s Law in Dielectrics Recall: 𝑄 𝑒𝑛𝑐𝑙 𝜖 0 = 𝐸 ∙ 𝑑𝐴 𝑄 𝑒𝑛𝑐𝑙 = 𝜎− 𝜎 𝑖 𝐴 Conductor Dielectrics 𝐸 =0 𝐸 ≠0 𝜎 −𝜎 𝑖 𝐸 ∙ 𝑑𝐴 =𝐸𝐴 𝐸𝐴= 𝜎− 𝜎 𝑖 𝐴 𝜖 0 𝜎 𝑖 =𝜎(1− 1 𝐾 ) A 𝐸𝐴= 𝜎𝐴 𝐾 𝜖 0 𝑄 𝑒𝑛𝑐𝑙−𝑓𝑟𝑒𝑒 𝜖 0 = 𝐾 𝐸 ∙ 𝑑𝐴

Gauss’s Law in Dielectrics Example: Capacitance of half filled spherical capacitor 𝑄 𝑒𝑛𝑐𝑙−𝑓𝑟𝑒𝑒 𝜖 0 = 𝐾 𝐸 ∙ 𝑑𝐴 K E1 𝑄 𝜖 0 = 𝐾 𝐸 ∙ 𝑑𝐴 =𝐾 𝐸 1 2𝜋 𝑟 2 + 𝐸 2 2𝜋 𝑟 2 ra 𝑄 1 𝜖 0 𝑄 2 𝜖 0 r rb 𝐸 1 = 𝑄 1 2𝜖 0 𝐾𝜋 𝑟 2 E2 𝐸 2 = 𝑄 2 2𝜖 0 𝜋 𝑟 2 𝑉= 𝑟 𝑎 𝑟 𝑏 𝐸 1 𝑑𝑟= 𝑄 1 ( 𝑟 𝑏 − 𝑟 𝑎 ) 2𝜖 0 𝐾𝜋 𝑟 𝑎 𝑟 𝑏 𝑄 1 = 2𝜖 0 𝐾𝜋 𝑟 𝑎 𝑟 𝑏 𝑉 ( 𝑟 𝑏 − 𝑟 𝑎 ) 𝑄= 𝑄 1 + 𝑄 2 = 2𝜖 0 𝜋 𝑟 𝑎 𝑟 𝑏 𝑉 𝑟 𝑏 − 𝑟 𝑎 (𝐾+1) 𝑉= 𝑟 𝑎 𝑟 𝑏 𝐸 2 𝑑𝑟= 𝑄 2 ( 𝑟 𝑏 − 𝑟 𝑎 ) 2𝜖 0 𝜋 𝑟 𝑎 𝑟 𝑏 𝑄 2 = 2𝜖 0 𝜋 𝑟 𝑎 𝑟 𝑏 𝑉 ( 𝑟 𝑏 − 𝑟 𝑎 ) 𝐶= 𝑄 𝑉 = 2𝜖 0 𝜋 𝑟 𝑎 𝑟 𝑏 (𝐾+1) ( 𝑟 𝑏 − 𝑟 𝑎 ) lim Κ→1 2𝜖 0 𝜋 𝑟 𝑎 𝑟 𝑏 (𝐾+1) ( 𝑟 𝑏 − 𝑟 𝑎 ) = 4𝜖 0 𝜋 𝑟 𝑎 𝑟 𝑏 ( 𝑟 𝑏 − 𝑟 𝑎 ) = C empty Check: K->1 needs to reproduce empty

Molecular Model of induced Charge

Clicker question A conductor is an extreme case of a dielectric, since if an electric field is applied to a conductor, charges are free to move within the conductor to set up “induced charges”. What is the dielectric constant of a perfect conductor? K = 0 K = A value depends on the material of the conductor