`-=[]⟨⟩\;',./~!@#$%^&*()_+{}|:"<>? 𝑎𝑏𝑐𝑑𝑒𝑓𝑔ℎ𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝𝑞𝑟𝑠𝑡𝑢𝑣𝑤𝑥𝑦𝑧 Å − × ⋅∓±∘꞊﹦∗∙ ℯ 𝔸𝔹ℂ𝔻𝔼𝔽𝔾ℍ𝕀𝕁𝕂𝕃𝕄ℕ𝕆ℙℚℝ𝕊𝕋𝕌𝕍𝕎𝕏𝕐ℤ𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻𝐼𝐽𝐾𝐿𝑀𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈𝑉𝑊𝑋𝑌𝑍 ∼∽∾≁≂≃≄≅≆≇≈≉≌≐≠≡ ≤≥≦≧≨≩≪≫ ∈∉∊∋∌∍ ⊂⊃⊄⊅⊆⊇ 𝛼𝛽𝛾𝛿𝜀𝜁𝜂𝜃𝜄𝜅𝜆𝜇𝜈𝜉𝜊𝜋𝜌𝜎𝜏𝜐𝜑𝜒𝜓𝜔 ∀∂∃∅⦰∆∇∎∞∝∴∵ ∏∐∑⋀⋁⋂⋃ ∧∨∩∪ ∫∬∭∮∯∰∱∲∳ ∥⋮⋯⋰⋱ ‖ ′ ″ ‴ ⁄ ⁗ ʹ ʺ ‵ ‶ ‷ ﹁﹂﹃﹄︹︺︻︼ ︗ ︘ ︿﹀︽︾﹇﹈︷︸ ⏜ ⏝ ⎴ ⎵ ⏞ ⏟ ⏠ ⏡ ←↑→↓↤↦↥↧↔↕↖↗↘↙▲▼◀▶↺↻⟲⟳ ↼↽↾↿⇀⇁⇂⇃⇄⇅⇆⇇ ⇐⇑⇒⇓⇔⇌⇍⇏⇕⇖⇗⇘⇙⇙⇳⥢⥣⥤⥥⥦⥧⥨⥩⥪⥫⥬⥭⥮⥯

Current Density and Conductivity

The current density 𝐽 is the current per unit area 𝐴,

𝐼=|𝑞|𝑛𝐴𝑣=|𝑞|𝑛𝐴𝑢𝐸

   𝐽≡𝐼𝐴=|𝑞|𝑛𝐴𝑢𝐸𝐴=|𝑞|𝑛𝑢𝐸

The conductivity 𝜎 is a mterial-dependent quantity. 𝐽=|𝑞|𝑛𝑢𝐸≡𝜎𝐸⇒𝜎=|𝑞|𝑛𝑢 Conductivity add, even for opposite sign charge carriers

That is 𝜎=|𝑞₊|𝑛₊𝑢₊+|𝑞₋|𝑛₋𝑢₋

Resistance

The resistance of a resistor of length 𝐿,

∆𝑉=−∫𝐸⋅𝑑𝑙⇒|∆𝑉|=𝐸𝐿

 𝐼=𝑞𝑛𝐴𝑢𝐸=𝜎𝐴𝐸

𝐼=𝜎𝐴𝐸=𝜎𝐴|∆𝑉|𝐿⇒|∆𝑉|=𝐼𝐿𝜎𝐴

∵|∆𝑉|=𝐼𝑅⇒𝑅=𝐿𝜎𝐴

Resistors in Series

When resistors are in series, the effective series resistance is

𝐼=𝐼₁=𝐼₂

        ∆𝑉_emf+∆𝑉₁+∆𝑉₂=0

⇒𝑉_emf−𝐼𝑅₁−𝐼𝑅₂=0

⇒𝑉_emf=𝐼𝑅₁+𝐼𝑅₂=𝐼(𝑅₁+𝑅₂)=𝐼𝑅_equivalent

⇒𝑅_equivalent=𝑅₁+𝑅₂

For resistor made of the same material and with the same 𝐴, the resistance follows straight from the definition of resistance 𝑅=𝐿𝜎𝐴

𝑅_equivalent=𝐿₁𝜎𝐴+𝐿₂𝜎𝐴=𝐿₁+𝐿₂𝜎𝐴=𝐿_equivalent𝜎𝐴

        ⇒𝐿_equivalent=𝐿₁+𝐿₂

Resistors in Parallel

When resistors are in parallel, the effective parallel resistance is

𝐼=𝐼₁+𝐼₂=𝑉_emf𝑅₁+𝑉_emf𝑅₂=1𝑅₁+1𝑅₂𝑉_emf=1𝑅_equivalent𝑉_emf

⇒1𝑅_equivalent=1𝑅₁+1𝑅₂=𝑅₁+𝑅₁𝑅₁𝑅₁

For resistor made of the same material and with the same 𝐿, the resistance follows straight from the definition of resistance 𝑅=𝐿𝜎𝐴

𝑅=𝐿𝜎𝐴⇒1𝑅=𝜎𝐴𝐿

⇒1𝑅_equivalent=𝜎𝐴₁𝐿+𝜎𝐴₁𝐿=𝜎𝐿(𝐴₁+𝐴₂)=𝜎𝐿𝐴_equivalent

        ⇒𝐴_equivalent=𝐴₁+𝐴₂

Capacitor in Series

When capacitors are in series, the effective series capacitance is

𝐼=𝐼₁=𝐼₂⇒𝑄₁=𝑄₂=𝑄

∆𝑉_emf+∆𝑉₁+∆𝑉₂=0

⇒𝑉_emf−𝑄𝐶₁−𝑄𝐶₂=0

⇒𝑉_emf=𝑄𝐶₁+𝑄𝐶₂=𝑄1𝐶₁+1𝐶₂=𝑄1𝐶_equivalent

⇒1𝐶_equivalent=1𝐶₁+1𝐶₂=𝐶₁+𝐶₂𝐶₁𝐶₂

For capacitor made of the same material and with the same 𝐴, the capacitance follows straight from the definition of capacitance 𝐶=𝜀₀𝐴𝑠

𝐶=𝜀₀𝐴𝑠⇒1𝐶=𝑠𝜀₀𝐴

⇒1𝐶_equivalent=𝑠₁𝜀₀𝐴+𝑠₂𝜀₀𝐴=1𝜀₀𝐴(𝑠₁+𝑠₂)=1𝜀₀𝐴𝑠_equivalent

⇒𝑠_equivalent=𝑠₁+𝑠₂

Capacitor in Parallel

When capacitors are in parallel, the effective parallel capacitance is

𝐼=𝐼₁+𝐼₂⇒𝑄=𝑄₁+𝑄₂

  ⇒𝐶_equivalent𝑉_emf=𝐶₁𝑉_emf+𝐶₂𝑉_emf

⇒𝐶_equivalent=𝐶₁+𝐶₂

For capacitor made of the same material and with the same 𝑠, the capacitance follows straight from the definition of capacitance 𝐶=𝜀₀𝐴𝑠

𝐶_equivalent=𝜀₀𝐴₁𝑠+𝜀₀𝐴₂𝑠=𝜀₀𝑠(𝐴₁+𝐴₂)=𝜀₀𝑠𝐴_equivalent

⇒𝐴_equivalent=𝐴₁+𝐴₂

Source and Reference

https://www.youtube.com/watch?v=C0hFkY2G4y4&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=18