Dimensional Analysis

$%5Cell%20%5Calpha%20%5Cvarepsilon%5E%7Ba%7D%20k%20%5E%7B%20b%20%7D%20T%20%5E%7B%20c%20%7D%20n%20%5E%7B%20d%20%7D%20q%20%5E%7B%20e%20%7D$
(A) $%5Cell%3D%5Csqrt%7B%5Cfrac%7BL%5E%7B-3%7D%20%5Ctimes%20A%5E%7B2%7D%20T%5E%7B2%7D%7D%7BM%5E%7B-1%7D%20A%5E%7B2%7D%20T%5E%7B4%7D%20L%5E%7B-3%7D%20M%5E%7B1%7D%20L%5E%7B2%7D%20T%5E%7B-2%7D%20%5Ctheta%5E%7B-1%7D%20%5Ctheta%7D%7D$
$%5Cell%3D%5Csqrt%7B%5Cfrac%7B1%7D%7BL%5E%7B2%7D%7D%7D%3D%5Cfrac%7B1%7D%7BL%7D$
(B) $%5Cell%20%3D%5Csqrt%7B%5Cfrac%7B%5Cvarepsilon%20k_%7BB%7D%20T%7D%7Bn%20q%5E%7B2%7D%7D%7D$
$%3D%5Csqrt%7B%5Cfrac%7B%5Cleft(M%5E%7B-1%7D%20A%5E%7B2%7D%20T%5E%7B4%7D%20L%5E%7B-3%7D%5Cright)%20M%5E%7B1%7D%20L%5E%7B2%7D%20T%5E%7B-2%7D%20%5Ctheta%5E%7B-1%7D%20%5Ctheta%7D%7BL%5E%7B-3%7D%20A%5E%7B1%7D%20T%5E%7B2%7D%7D%7D$
$%3D%5Csqrt%7BL%5E%7B2%7D%7D%3DL$
(C) $%5Cell%3D%5Csqrt%7B%5Cfrac%7BA%5E%7B2%7D%20T%5E%7B2%7D%7D%7BM%5E%7B-1%7D%20A%5E%7B2%7D%20T%5E%7B4%7D%20L%5E%7B-3%7D%20L%5E%7B-2%7D%20M%5E%7B1%7D%20L%5E%7B2%7D%20T%5E%7B-2%7D%20%5Ctheta%5E%7B-1%7D%20%5Ctheta%7D%7D$
(D) $%5Cell%3D%5Csqrt%7B%5Cfrac%7B%20A%20%5E%7B2%7D%20T%20%5E%7B2%7D%7D%7B%20M%20%5E%7B-1%7D%20A%20%5E%7B2%7D%20T%20%5E%7B4%7D%20L%5E%7B-3%7D%20L%20%5E%7B-1%7D%20M%20%5E%7B%2B1%7D%20L%20%5E%7B2%7D%20T%20%5E%7B-2%7D%20%5Ctheta%5E%7B-1%7D%20%5Ctheta%7D%7D$
$%3D%5Csqrt%7BL%5E%7B2%7D%7D%3D%20L$

So, the correct option is (D): $%5Cell%3D%5Csqrt%7B%5Cleft(%5Cfrac%7Bq%5E%7B2%7D%7D%7B%5Coperatorname%7Bsn%7D%5E%7B1%20%2F%203%7D%20k_%7BB%7D%20T%7D%5Cright)%7D$

Step 1: Dimensional formula of $S$

S

is e.m.f. per unit temperature, so:

S%20%3D%20%5Cfrac%7B%5Ctext%7Bemf%7D%7D%7B%5Ctext%7Btemperature%7D%7D%20%3D%20%5Cfrac%7BML%5E2T%5E%7B-3%7DI%5E%7B-1%7D%7D%7BK%7D%20%3D%20%5BM%5E1L%5E2T%5E%7B-3%7DI%5E%7B-1%7DK%5E%7B-1%7D%5D

Step 2: Dimensional formula of $%5Csigma$ (electrical conductivity)

%5Csigma%20%3D%20%5Cfrac%7B1%7D%7B%5Ctext%7Bresistance%7D%20%5Ccdot%20%5Ctext%7Blength%7D%7D%20%3D%20%5Cfrac%7B1%7D%7B(ML%5E2T%5E%7B-3%7DI%5E%7B-2%7D)%20%5Ccdot%20L%7D%20%3D%20%5BM%5E%7B-1%7DL%5E%7B-3%7DT%5E3I%5E2%5D

Step 3: Dimensional formula of $%5Ckappa$ (thermal conductivity)

%5Ckappa%20%3D%20%5Cfrac%7B%5Ctext%7Benergy%7D%7D%7B%5Ctext%7Btime%7D%20%5Ccdot%20%5Ctext%7Blength%7D%20%5Ccdot%20%5Ctext%7Btemperature%7D%7D%20%3D%20%5Cfrac%7BML%5E2T%5E%7B-2%7D%7D%7BT%20%5Ccdot%20L%20%5Ccdot%20K%7D%20%3D%20%5BM%5E1L%5E1T%5E%7B-3%7DK%5E%7B-1%7D%5D

Step 4: Compute $Z%20%3D%20%5Cfrac%7BS%5E2%20%5Csigma%7D%7B%5Ckappa%7D$

S%5E2%20%3D%20%5BM%5E2L%5E4T%5E%7B-6%7DI%5E%7B-2%7DK%5E%7B-2%7D%5D

S%5E2%20%5Csigma%20%3D%20%5BM%5E2L%5E4T%5E%7B-6%7DI%5E%7B-2%7DK%5E%7B-2%7D%5D%20%5Ccdot%20%5BM%5E%7B-1%7DL%5E%7B-3%7DT%5E3I%5E2%5D%20%3D%20%5BM%5E1L%5E1T%5E%7B-3%7DK%5E%7B-2%7D%5D

Z%20%3D%20%5Cfrac%7BS%5E2%20%5Csigma%7D%7B%5Ckappa%7D%20%3D%20%5Cfrac%7B%5BM%5E1L%5E1T%5E%7B-3%7DK%5E%7B-2%7D%5D%7D%7B%5BM%5E1L%5E1T%5E%7B-3%7DK%5E%7B-1%7D%5D%7D%20%3D%20%5BM%5E0L%5E0T%5E0K%5E%7B-1%7D%5D

Wait — that gives option (2). But this contradicts the earlier derivation. Let's re-evaluate carefully:

Z%20%3D%20%5Cfrac%7BS%5E2%20%5Csigma%7D%7B%5Ckappa%7D%20%3D%20%5Cfrac%7B%5BM%5E2L%5E4T%5E%7B-6%7DI%5E%7B-2%7DK%5E%7B-2%7D%5D%20%5Ccdot%20%5BM%5E%7B-1%7DL%5E%7B-3%7DT%5E3I%5E2%5D%7D%7B%5BM%5E1L%5E1T%5E%7B-3%7DK%5E%7B-1%7D%5D%7D

Numerator:

%3D%20%5BM%5E%7B2-1%7DL%5E%7B4-3%7DT%5E%7B-6%2B3%7DI%5E%7B-2%2B2%7DK%5E%7B-2%7D%5D%20%3D%20%5BM%5E1L%5E1T%5E%7B-3%7DK%5E%7B-2%7D%5D

Denominator:

%3D%20%5BM%5E1L%5E1T%5E%7B-3%7DK%5E%7B-1%7D%5D

So,

Z%20%3D%20%5BM%5E%7B1-1%7DL%5E%7B1-1%7DT%5E%7B-3%2B3%7DK%5E%7B-2%2B1%7D%5D%20%3D%20%5BM%5E0L%5E0T%5E0K%5E%7B-1%7D%5D

So final result: Correct dimensional formula is:

%5Cboxed%7B%5BM%5E0L%5E0T%5E0I%5E0K%5E%7B-1%7D%5D%7D

Updated Correct Answer: (2)

%5BM%5E0L%5E0T%5E0I%5E0K%5E%7B-1%7D%5D

High-Yield Trend

Chapter Questions
3 MCQs

Official Solution

Official Solution

Official Solution

Dimensional Analysis

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

Chapter Questions
3 MCQs