Kirchoff's Rules

Junction Rule:

                 \(i_1\ +\ i_2\ +\ i_4=i_3\ +\ i_5\)

At any junction, the sum of current's entering the junction is equal to the sum of current leaving the junction.

Loop Rule:

The algebraic sum of all the potential difference along a closed loop is zero.

         \(\Sigma\ E\ =\ 0\)

Grouping of Cell

Parallel Grouping:

\(Eeq\ =\ \frac{E_1\ r_2\ +\ E_2\ r_1}{r_1\ +\ r_2}\)

\(i\ =\ \frac{Eeq}{R\ +\ \frac{r_1\ r_2}{r_1\ +\ r_2}}\)

Series Grouping:

    \(i=\ \frac{E_1\ +\ E_2}{R_1\ +\ (r_1\ +\ r_2)}\ =\ \frac{E_0}{R_1\ +\ R_2}\)

\(E_0=\ E_1\ +\ E_2,\ R_2\ =\ r_1\ +\ r_2\)

Field due to a current carrying circular ring

\(\bullet\)  Field at a point far away from the centre

                   For   \(d\ \gt\ \gt\ \gt a\)

                          \(B=\ \frac{\mu_0 \ i\ a^2}{2\ d^3}\)

\(\bullet\)  Field at an axial point

                       \(B=\ \frac{\mu_0 \ i\ a^2}{2(a^2\ +\ d^2)^{3/2}}\)

\(\bullet\)  Field at the centre

                  \(B=\ \frac{\mu_0\ i}{2a}\)

Force on a current carrying conductor

\(d\ \vec{F}=\ i(\vec{dl}\ \times\ \vec B)\ F=\ i\ l\ B\)

Ampere's Law

\(\oint\ \overline B\ \cdot\ \overline {dl}=\ \mu_0\ i\)

Where i = Total current crossing the area bounded by closed curve.

Electric Cell

It is a device which maintains a potential difference between its two terminals.

EMF

\(emf\ =\ \frac{w}{q}\)

It is equal to the potential difference between the terminals when they are not connected externally.

Internal Resistance

         \(V_A\ -\ V_B\ =\ E\ -\ i\ r\)

          r =  Internal resistance of the cell

Wheatstone Bridge:

Condition for balanced Wheatstone bridge

If \(\frac{R_1}{R_2}\ =\ \frac{R_3}{R_4}\)

    \(V_c\ =\ V_d\)

No current through G

    \(I_g\ \gt\ 0\)

Electric Current

Rate of flow of charge through an area

   \(i= \ \frac{Q}{t}\)

   \(i\ =\ \int\ \vec J\ \cdot\ \vec{ds}\)

Current Density

\(\vec J=\ \frac{\Delta\ i}{\Delta\ s\ \cos\ \theta}\ \sigma\ E\)

\(\Delta S=\ Area\ of\ cross-section\)

\(\theta=\) Angle between Area Vector and current flow

\(\sigma=\ conductivity\)

Drift Velocity

\(V_d=\ \frac{i}{n_e\ A}=\ \frac{J}{n_e}\)

Mobility

\(\mu=\ \frac{V_d}{E}=\ \frac{e\ T}{m}\)

T =  Average collision time

Ohm's Law

\(J=\ \sigma\ E\ \ or\ \ V=\ I\ R\)

Resistivity

\(e=\ \frac{1}{\sigma}\ \Omega\ m\)

Conductivity

\(\sigma= \ J/E\)

Grouping of Resistors

In series combination:

\(Req=\ R_1\ +\ R_2\ +\ R_3\ +\ ...\ +\ R_n\)

For two resistors,

\(Req=\ R_1\ +\ R_2\)

In parallel combination

\(\frac{1}{Req}=\ \frac{1}{R_1}\ +\ \frac{1}{R_2}\ +\ ...\ +\ \frac{1}{R_n}\)

For two resistors

\(Req=\ \frac{R_1\ R_2}{R_1\ +\ R_2}\)

Temperature Dependence of Resistance

\(R= \ R_0\ [1\ +\ \alpha\ (T\ -\ T_0)]\)

\(e= \ e_0\ [1\ +\ \alpha\ (T\ -\ T_0)]\)

R =  Resistance at temperature T

\(R_0=\) Resistance at temperature \(T_0\)

\(\alpha=\)  Temperature coefficient of resistance.