Question

The current flowing through a resistor R varies as I= kt , where k is constant. Find the rms current for first 2 seconds.

Answer 1

RMS CURRENT for t = 2 sec :

$\: i_{RMS} = \sqrt{ \frac{ \displaystyle\int_{0}^{2} {i}^{2}dt }{ \displaystyle\int_{0}^{2}dt} }$

$\implies i_{RMS} = \sqrt{ \frac{ \displaystyle\int_{0}^{2} {(kt)}^{2}dt }{ \displaystyle\int_{0}^{2}dt} }$

$\implies i_{RMS} =k \sqrt{ \frac{ \displaystyle\int_{0}^{2} {t}^{2} dt }{ \displaystyle\int_{0}^{2}dt} }$

$\implies i_{RMS} =k \sqrt{ \dfrac{ \displaystyle \frac{ {t}^{3} }{3} \bigg | _{0}^{2} }{ 2 - 0} }$

$\implies i_{RMS} =k \sqrt{ \dfrac{ \displaystyle (\frac{ 8}{3}) }{ 2 } }$

$\implies i_{RMS} =k \sqrt{ \dfrac{4 }{3}}$

$\boxed{ \implies i_{RMS} = \dfrac{2k}{ \sqrt{3} } \: units}$

Hope It Helps.

Answer 2

Explanation:

RMS CURRENT for t = 2 sec :

\: i_{RMS} = \sqrt{ \frac{ \displaystyle\int_{0}^{2} {i}^{2}dt }{ \displaystyle\int_{0}^{2}dt} }i

RMS

=

∫

0

2

dt

∫

0

2

i

2

dt

\implies i_{RMS} = \sqrt{ \frac{ \displaystyle\int_{0}^{2} {(kt)}^{2}dt }{ \displaystyle\int_{0}^{2}dt} }⟹i

RMS

=

∫

0

2

dt

∫

0

2

(kt)

2

dt

\implies i_{RMS} =k \sqrt{ \frac{ \displaystyle\int_{0}^{2} {t}^{2} dt }{ \displaystyle\int_{0}^{2}dt} }⟹i

RMS

=k

∫

0

2

dt

∫

0

2

t

2

dt

\implies i_{RMS} =k \sqrt{ \dfrac{ \displaystyle \frac{ {t}^{3} }{3} \bigg | _{0}^{2} }{ 2 - 0} }⟹i

RMS

=k

2−0

3

t

3

∣

∣

∣

∣

∣

0

2

\implies i_{RMS} =k \sqrt{ \dfrac{ \displaystyle (\frac{ 8}{3}) }{ 2 } }⟹i

RMS

=k

2

(

3

8

)

\implies i_{RMS} =k \sqrt{ \dfrac{4 }{3}}⟹i

RMS

=k

3

4

\boxed{ \implies i_{RMS} = \dfrac{2k}{ \sqrt{3} } \: units}

⟹i

RMS

=

3

2k

units

Hope It Helps.