Answer:
[tex]\tau=3.3*10^{-6}s[/tex]
Explanation:
Take at look to the picture I attached you, using Kirchhoff's current law we get:
[tex]C*\frac{dV}{dt}+\frac{V}{R}=0[/tex]
This is a separable first order differential equation, let's solve it step by step:
Express the equation this way:
[tex]\frac{dV}{V}=-\frac{1}{RC}dt[/tex]
integrate both sides, the left side will be integrated from an initial voltage v to a final voltage V, and the right side from an initial time 0 to a final time t:
[tex]\int\limits^V_v {\frac{dV}{V} } =-\int\limits^t_0 {\frac{1}{RC} } \, dt[/tex]
Evaluating the integrals:
[tex]ln(\frac{V}{v})=e^{\frac{-t}{RC} }[/tex]
natural logarithm to both sides in order to isolate V:
[tex]V(t)=ve^{-\frac{t}{RC} }[/tex]
Where the term RC is called time constant and is given by:
[tex]\tau=R*C=10*(0.330*10^{-6})=3.3*10^{-6}s[/tex]
