D'après la loi d'additivité des tensions, ![{\displaystyle E=u_{R}+u_{c}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fca7b253e621597887813f2a2163cb885d98e90)
on a ![{\displaystyle u_{R}=Ri\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67010489caecede6a26f0564cccaf4c6754ec980)
d'où ![{\displaystyle E=Ri+u_{c}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2726810b4be1c2c0cac15cf0206d7fed873fc2)
on a
d'où ![{\displaystyle i={\frac {dq}{dt}}={\frac {d(Cu_{c})}{dt}}=C{\frac {du_{c}}{dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9995b406d72d5d90a48a468ea6b3128b0fc8e3a)
Equation différentielle du circuit RC :
La solution de cette équation différentielle est de la forme ![{\displaystyle u_{c}(t)=Ae^{-kt}+B\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20dbf51720d5e161a879b2fc4a8219b2d5ab8a46)
quand ![{\displaystyle t\rightarrow \infty ,u_{c}=B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4faf42d5272399138f19302f37142194db8bc993)
![{\displaystyle {\frac {d}{dt}}(Ae^{-kt}+B)+{\frac {1}{RC}}(Ae^{-kt}+B)={\frac {E}{RC}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06ea0558abaad0db4f22e0e832361177d3d42f49)
![{\displaystyle -kAe^{-kt}+{\frac {1}{RC}}(Ae^{-kt}+B)={\frac {E}{RC}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6992d614cef0752b9452f3721a8d90b6c6ae260)
![{\displaystyle Ae^{-kt}(-m+{\frac {1}{RC}})+{\frac {1}{RC}}B={\frac {E}{RC}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89723455b0a3d5cc418c8c88f4cbb8125ef230f3)
![{\displaystyle \forall t,{\begin{cases}-m+{\frac {1}{RC}}=0\\B=E\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2124f0f6280114b9a2ad120b9b8194151d14e5a0)
d'où ![{\displaystyle k={\frac {1}{RC}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cdb0cd0686084f1f46951019c411a5846ad04a)
Les conditions initiales donnent A :
à
d'où ![{\displaystyle A=-E\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32e0eac2115f7908a567bfa48da6ae9287d649ed)
Solution de l'équation différentielle :
![{\displaystyle u_{c}(t)=-Ee^{-kt}+E\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cef40272a5aecdf6b3e391055b0b570324be5df2)
soit <math>u_c(t) = E(1-e^{-\frac{t}{RC}})