If the sum of first 4 terms of an AP is 6 and sum of first 6 terms is 4, then sum of first 12 terms of AP is :
Show Hint
When given sums of different numbers of terms in an AP, you can set up a system of linear equations for the first term 'a' and common difference 'd'. Solving this system is the key to finding any other property of the AP.
Concept:
Magnetic energy density \(u\) inside a solenoid is given by \(u = \frac{B^2}{2\mu_0}\). The magnetic field \(B\) is related to current by \(B = \mu_0 n I\).
Step 1: Determine the specific current value.
Max current \(I_0 = \frac{\varepsilon}{R} = \frac{10}{10} = 1\,\text{A}\).
The problem asks for the instant when \(I = \frac{I_0}{e} = \frac{1}{e}\,\text{A}\).
Step 2: Calculate the magnetic field B.
Using \(n = 10^4\) turns/m:
\[
B = \mu_0 n I = (4\pi \times 10^{-7}) (10^4) \left(\frac{1}{e}\right) = \frac{4\pi \times 10^{-3}}{e}
\]
Step 3: Calculate energy density.
\[
u = \frac{B^2}{2\mu_0} = \frac{1}{2\mu_0} \left( \frac{4\pi \times 10^{-3}}{e} \right)^2
\]
\[
u = \frac{16\pi^2 \times 10^{-6}}{2(4\pi \times 10^{-7}) e^2} = \frac{16\pi^2}{8\pi} \frac{10}{e^2} = \frac{20\pi}{e^2}
\]
Step 4: Match with the given form.
Given form is \(\alpha \frac{\pi}{e^2}\).
Comparing: \(\alpha = 20\).
\[
\boxed{\alpha = 20}
\]
