The emf (\( \varepsilon \)) induced in an inductor is related to the rate of change of current and self-inductance (\( L \)) by the formula:
\[ |\varepsilon| = L \frac{dI}{dt}. \]
Step 1: Determine the rate of change of current
The given current is:
\[ I = 3t + 8. \]
Differentiate \( I \) with respect to \( t \):
\[ \frac{dI}{dt} = 3 \, \text{A/s}. \]
Step 2: Substitute the given values
The magnitude of induced emf is \( |\varepsilon| = 12 \, \text{mV} = 12 \times 10^{-3} \, \text{V} \). Substituting into the formula:
\[ 12 \times 10^{-3} = L \cdot 3. \]
Step 3: Solve for \( L \)
Rearrange to find \( L \):
\[ L = \frac{12 \times 10^{-3}}{3} = 4 \times 10^{-3} \, \text{H}. \]
Convert to millihenries:
\[ L = 4 \, \text{mH}. \]
Thus, the self-inductance of the inductor is \( L = 4 \, \text{mH} \).
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32