Problem 15.82 Challenge

a) พิจารณาวงจร RC แบบชาร์จประจุไฟฟ้า ที่ \(t = 2RC\)

\(_{-t/τ}\)

จาก \(q = Q(1 - e)\)

\(_{-2RC/RC}\)

\(q = εC(1 - e)\)

\(_{-2}\)

\(q=εC(1-e)\)

\(_{-2}\)

∴ในวงจรLCจะได้\(Q=εC(1-e)=0.867εC\;\blacksquare\)

b) จาก \(I = ω_{0}Q\)

\(I =\frac{Q}{LC}; \omega _{0} =\frac{1}{LC}\)

\(I =\frac{0.867εC}{LC}\)

\(I = 0.867ε\sqrt{\frac{C}{L}}\;\blacksquare\)

c) จาก \(U_{C}=\frac{q^{2}}{2C}\)

\(U_{C}=\frac{(Qcos(\omega _{0}t))^{2}}{2C}\)

\(U_{C}=\frac{(0.867εC)^{2}\cos ^{2}(\frac{t}{LC})}{2C}\)

\(U_{C}= 0.376ε^{2}Ccos^{2}(\frac{t}{LC}) \;\blacksquare\)

d) จาก \(U_{L}=\frac{Li^{2}}{2}\)

\(U_{L}=\frac{L(-Isin(\omega _{0}t))^{2}}{2}\)

\(U_{L}=\frac{L(-0.867ε\frac{C}{L})^{2}\sin ^{2}(\frac{t}{LC})}{2}\)

\(U_{L}= 0.376ε^{2}Csin^{2}(\frac{t}{LC}) \;\blacksquare\)

e) จาก \(U = U_{C}+ U_{L}\)

\(U = 0.376ε^{2}Ccos^{2}(\frac{t}{LC}) + 0.376ε^{2}Csin^{2}(\frac{t}{LC})\)

\(U = 0.376ε^{2}C(\cos ^{2}(\frac{t}{LC}) + \sin ^{2}(\frac{t}{LC}))\)

\(U = 0.376ε^{2}C \;\blacksquare\)