Question : What is the dimensional formula for RC, LC, L/R? Where R is Resistance, C is capacitance and L is inductance.
Solution :
Dimension of RC
We know
Q=CV
Where
Q = Charge
C = Capacitance
V = Potnetial
Q=CV
Where
Q = Charge
C = Capacitance
V = Potnetial
Q=CV
Q=C(IR) [∵ V=IR]
Q=CIR
Q=C(Q/t)R [∵ I=Q/t]
Qt/Q = RC
Q=CIR
Q=C(Q/t)R [∵ I=Q/t]
Qt/Q = RC
t = RC
RC = Time
So, the dimensional formula of RC must be equal to the dimensions of Time.
Hence, dimensions of RC is [M0L0T1]
RC = Time
So, the dimensional formula of RC must be equal to the dimensions of Time.
Hence, dimensions of RC is [M0L0T1]
Dimension of LC
We know,
Resonance frequency is given by
f = 1/2π√LC
√LC = 1/2πf
Resonance frequency is given by
f = 1/2π√LC
√LC = 1/2πf
√LC = T/2π [ ∵ 1/f = T]
Squaring both sides
LC = T2/4π
4π don't have any dimension.
Hence, the dimensional formula of LC is [M0L0T2]
Squaring both sides
LC = T2/4π
4π don't have any dimension.
Hence, the dimensional formula of LC is [M0L0T2]
Dimension of L/R
We know
ε = - LdI/dt
|ε| = LdI/dt
V = LdI/dt
[∵ EMF (ε) and Voltage (V) are having the same dimensions]
|ε| = LdI/dt
V = LdI/dt
[∵ EMF (ε) and Voltage (V) are having the same dimensions]
IR = LdI/dt [∵ V=IR]
Idt/dI = L/R
Idt/dI = L/R
L/R = Idt/dI
I and dI have the same dimensions so they will cancel out each other and we can say that L/R will have the dimensions of Time.
Hence, the dimension of L/R is [M0L0T1]
