question_answer

If C is capacitance, and R is resistance, then calculate the possible units of \[\text{C}{{\text{R}}^{\text{2}}}\].

Answer:

                It is known that time constant of RC circuit is                                                                                                  \[\tau =\text{RC}\]  and time constant of LR circuit is                                                                                           \[\tau =\text{L}/\text{R}\]                                                                                                       \[\therefore\]                                                                                                            \[\text{RC =}\frac{L}{R}\]  or                                                                          \[\text{C}{{\text{R}}^{\text{2}}}=\text{L}\] Hence the units of                                                                                   \[\text{C}{{\text{R}}^{\text{2}}}\] are those of inductance (L) viz. henry From                                                                             \[e=L\frac{dI}{dt},\,\,\,L=\frac{dt}{dI}\]                                                                                                                                                       \[\therefore\]             Units of L are also                                                                         \[\frac{Volt-\,\,\sec }{ampere}=ohm-\sec\] . Again, from                                                                                            \[E=\frac{1}{2}L{{I}^{2}}\] ,                                                                                             \[L=\frac{2E}{{{I}^{2}}}\] Therefore, units of L are also                                                                 \[\text{joule}/\text{amper}{{\text{e}}^{\text{2}}}\] .