Answer
434.1k+ views
Hint: The net magnetic field is the sum from each individual loop and is maximum in the middle of the solenoid because that point minimizes the average distance to each loop
Complete step-by-step solution:
Given,
Inductance, \[L=10\text{ H}\]
Resistance, \[r=2\text{ ohm}\]
Volt, \[V=10\text{ V}\]
Since, \[V=iR\]; where i is current
Therefore, current, \[{{i}_{o}}=\dfrac{10}{2}=5\text{A}\]
Now Maximum energy is,
\[\begin{gathered}
& {{E}_{0}}=\dfrac{1}{2}L\times {{i}_{o}} \\
& \text{ }=\dfrac{1}{2}\left( 10 \right)\times {{\left( 5 \right)}^{2}} \\
& \text{ }=125\text{ J} \\
\end{gathered}\]
Thus. \[1/4\text{th}\]of the total energy is,
\[\begin{gathered}
& E=\dfrac{1}{4}{{E}_{0}} \\
& \text{ }=\dfrac{{{E}_{0}}}{4} \\
& \text{ }=\dfrac{125}{4}\text{ J} \\
\end{gathered}\]
When \[E=\dfrac{125}{4}\text{ J}\], assume the current to be \[i\]
Now,
Current \[i\] is,
\[\begin{gathered}
& \text{ }E=\dfrac{1}{2}\times L\times {{i}^{2}} \\
& \Rightarrow \dfrac{125}{4}=\dfrac{1}{2}\times \left( 10 \right)\times {{i}^{2}} \\
& \Rightarrow \dfrac{125}{4}=5\times {{i}^{2}} \\
& \Rightarrow i=\dfrac{5}{2} \\
& \Rightarrow i=2.5\text{ A} \\
\end{gathered}\]
Now time taken to rise from 0 A to \[2.5\text{ A}\],
\[\begin{gathered}
& \text{ }i={{i}_{o}}\left[ 1-{{e}^{-Rt/L}} \right] \\
& \Rightarrow 2.5=5\left[ 1-{{e}^{-Rt/L}} \right] \\
& \Rightarrow {{e}^{-Rt/L}}=0.5 \\
& \Rightarrow \dfrac{-Rt}{L}={{\log }_{e}}\left( 0.5 \right) \\
& \Rightarrow \dfrac{Rt}{L}=0.693 \\
& \Rightarrow t=\dfrac{6.93}{2} \\
& \Rightarrow t=3.46\,\approx 3.5\text{ sec} \\
\end{gathered}\]
Hence, the correct answer is option A.
Note: while solving the formula, \[i={{i}_{o}}\left[ 1-{{e}^{-Rt/L}} \right]\] make sure that the exponential function is converted to logarithm function to make the solution easier to solve.
Complete step-by-step solution:
Given,
Inductance, \[L=10\text{ H}\]
Resistance, \[r=2\text{ ohm}\]
Volt, \[V=10\text{ V}\]
Since, \[V=iR\]; where i is current
Therefore, current, \[{{i}_{o}}=\dfrac{10}{2}=5\text{A}\]
Now Maximum energy is,
\[\begin{gathered}
& {{E}_{0}}=\dfrac{1}{2}L\times {{i}_{o}} \\
& \text{ }=\dfrac{1}{2}\left( 10 \right)\times {{\left( 5 \right)}^{2}} \\
& \text{ }=125\text{ J} \\
\end{gathered}\]
Thus. \[1/4\text{th}\]of the total energy is,
\[\begin{gathered}
& E=\dfrac{1}{4}{{E}_{0}} \\
& \text{ }=\dfrac{{{E}_{0}}}{4} \\
& \text{ }=\dfrac{125}{4}\text{ J} \\
\end{gathered}\]
When \[E=\dfrac{125}{4}\text{ J}\], assume the current to be \[i\]
Now,
Current \[i\] is,
\[\begin{gathered}
& \text{ }E=\dfrac{1}{2}\times L\times {{i}^{2}} \\
& \Rightarrow \dfrac{125}{4}=\dfrac{1}{2}\times \left( 10 \right)\times {{i}^{2}} \\
& \Rightarrow \dfrac{125}{4}=5\times {{i}^{2}} \\
& \Rightarrow i=\dfrac{5}{2} \\
& \Rightarrow i=2.5\text{ A} \\
\end{gathered}\]
Now time taken to rise from 0 A to \[2.5\text{ A}\],
\[\begin{gathered}
& \text{ }i={{i}_{o}}\left[ 1-{{e}^{-Rt/L}} \right] \\
& \Rightarrow 2.5=5\left[ 1-{{e}^{-Rt/L}} \right] \\
& \Rightarrow {{e}^{-Rt/L}}=0.5 \\
& \Rightarrow \dfrac{-Rt}{L}={{\log }_{e}}\left( 0.5 \right) \\
& \Rightarrow \dfrac{Rt}{L}=0.693 \\
& \Rightarrow t=\dfrac{6.93}{2} \\
& \Rightarrow t=3.46\,\approx 3.5\text{ sec} \\
\end{gathered}\]
Hence, the correct answer is option A.
Note: while solving the formula, \[i={{i}_{o}}\left[ 1-{{e}^{-Rt/L}} \right]\] make sure that the exponential function is converted to logarithm function to make the solution easier to solve.
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