[A⁻¹]
[LT²]
[LTA]
[LA⁻¹]
The correct option is C
[L]=[ML²T⁻²A⁻²]
[r]=[ML²T⁻³A⁻²]
[c]=[M⁻¹L⁻²T⁴A²]
[v]=[ML²T⁻³A⁻¹]
Therefore, the dimension of Lrcv is:
[Lrcv] = [ML²T⁻²A⁻²][ML²T⁻³A⁻²][M⁻¹L⁻²T⁴A²][ML²T⁻³A⁻¹]
= [M¹L²T⁻⁵A⁻⁴][M⁻¹L⁻²T⁴A²][ML²T⁻³A⁻¹]
= [M¹⁻¹⁺¹L²⁻²⁺²T⁻⁵⁺⁴⁻³A⁻⁴⁺²⁻¹]
= [M¹L²T⁻⁴A⁻³]
However, this is not among the options. Let's recalculate with the correct dimensions:
[L] = [M L² T⁻² A⁻²]
[R] = [M L² T⁻³ A⁻²]
[C] = [M⁻¹ L⁻² T⁴ A²]
[V] = [M L² T⁻³ A⁻¹]
Therefore,
[LRCV] = [M L² T⁻² A⁻²] [M L² T⁻³ A⁻²] [M⁻¹ L⁻² T⁴ A²] [M L² T⁻³ A⁻¹]
= [M¹⁺¹⁻¹⁺¹ L²⁺²⁻²⁺² T⁻²⁻³⁺⁴⁻³ A⁻²⁻²⁺²⁻¹]
= [M¹ L² T⁻⁴ A⁻³]
Let's check the dimensions of the options:
A) [A⁻¹]
B) [LT²]
C) [LTA]
D) [LA⁻¹]
Let's consider the formula for the time constant of an LR circuit: τ = L/R
[τ] = [T]
[L/R] = [L]/[R] = [ML²T⁻²A⁻²]/[ML²T⁻³A⁻²] = [T]
For an RC circuit: τ = RC
[τ] = [T]
[RC] = [M⁻¹L⁻²T⁴A²][ML²T⁻³A⁻²] = [T]
For an LC circuit: ω = 1/√LC
[ω] = [T⁻¹]
[1/√LC] = [M⁻¹L⁻²T⁴A²]⁻¹/²[ML²T⁻²A⁻²]⁻¹/² = [M⁰L⁰T⁻¹A⁰] = [T⁻¹]
Let's check the dimensions of LRCV:
[L] = Henry = [ML²T⁻²A⁻²]
[R] = Ohm = [ML²T⁻³A⁻²]
[C] = Farad = [M⁻¹L⁻²T⁴A²]
[V] = Volt = [ML²T⁻³A⁻¹]
[LRCV] = [M¹L²T⁻⁴A⁻³]
None of the options match. There might be an error in the question or options.