Input
output
Ideal Op-Amp Assumptions
1. No Current Flows into the Input Terminals
2.The Differential Input Voltage is Zero as V1 = V2 = 0 (Virtual Earth)
Let i be the current flowing through the resistors
$$ i = \frac{ V_{in} - V_{out} }{ R_{in} + R_{f}} $$
$$ i = \frac{V_{in} - V_{2}}{R_{in}} = \frac{V_{2} - V_{out}}{R_{in}} $$
$$ i = \frac{V_{in}}{R_{in}} - \frac{V_{2}}{R_{in}} = \frac{V_{2}}{R_{f}} - \frac{V_{out}}{R_{f}} $$
$$ \frac{V_{in}}{R_{in}} = V_{2}\left[ \frac{1}{R_{in}} + \frac{1}{R_{f}} \right] - \frac{V_{out}}{R_{f}} $$
As per ideal Op-Amp Assumption, V2 = 0
$$ \frac{V_{in}}{R_{in}} = 0\left[ \frac{1}{R_{in}} + \frac{1}{R_{f}} \right] - \frac{V_{out}}{R_{f}} $$
$$ \frac{V_{in}}{R_{in}} = - \frac{V_{out}}{R_{f}} $$
$$ V_{out} = -V_{in} .\frac{R_{f}}{R_{in}} $$
$$ V_{out} = V_{in} . A_{v} $$
where
\[ \boxed { \text{Output Voltage}(V_{out}) = A_{v} . V_{in} } \]
\[ \boxed { \text{Voltage gain}( A_{v}) = - \frac{R_{f}}{R_{in}} } \]