A length –scale (L) depends on the permittivity ε of a dielectric material ,Boltzmann constant kB , the absolute temperature (T) , the number per unit volume (n) of certain charged particles , and the charge (q) carried by each of the particles. Which of the following $$expression(s)$$ for L is (are) dimensionally correct?

Question

A length –scale (L) depends on the permittivity ε  of a dielectric material ,Boltzmann constant kB , the absolute temperature (T) , the number per unit volume (n) of certain charged particles , and the charge (q) carried by each of the particles. Which of the following $$expression(s)$$ for L is (are) dimensionally correct?

Infinity Learn · Accepted Answer

Permittivity  ε$$=M$$−$$1L$$−$$3T4I2Boltzmen$$ constant  $$KB=M1L2T$$−$$2K$$−$$1Temperature$$ T $$=$$ KNo.of particles per unit volume $$n=L$$−$$3Charge$$  $$q=I1T1For$$ A:  $$nq2$$ε$$KBT=L$$−$$3I2T2M$$−$$1L$$−$$3T4I2M1L2T$$−$$2K$$−$$1K=1L=L$$−$$1For$$ B: ε$$KBTnq2=L$$    [ from option A]For C: $$q2$$ε$$n2/3KBT=I2T2M$$−$$1L$$−$$3T4I2L$$−$$2ML2T$$−$$2K$$−$$1K=L3/2$$ For D:  $$q2$$ε$$n1/3KBT=I2T2M$$−$$1L$$−$$3T4I2L$$−$$1ML2T$$−$$2K$$−$$1K=L$$

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