The magnetic field inside a solenoid is given by:
\[ B = \mu_0 n i, \]
where:
- \( B = 6.28 \times 10^{-3} \, \text{T} \) is the magnetic field,
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \) is the permeability of free space,
- \( n = \frac{m}{\ell} \) is the number of turns per unit length,
- \( i = 5 \, \text{A} \) is the current,
- \( \ell = 0.5 \, \text{m} \) is the length of the solenoid.
Step 1: Rearranging the Formula
Substituting the given values:
\[ \mu_0 \left( \frac{m}{\ell} \right) i = B. \]
Rearranging to find \( m \):
\[ m = \frac{B \ell}{\mu_0 i}. \]
Step 2: Substituting the Values
Substituting the given values:
\[ m = \frac{6.28 \times 10^{-3} \times 0.5}{4\pi \times 10^{-7} \times 5}. \]
Simplifying:
\[ m = \frac{6.28 \times 10^{-3} \times 0.5}{12.56 \times 10^{-7}}. \]
Further simplification:
\[ m = \frac{3.14 \times 10^{-3}}{12.56 \times 10^{-7}}. \]
Calculating:
\[ m = 500. \]
Therefore, the value of \( m \) is 500.
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