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.
In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of $\mathrm{M}^{\mathrm{B}} \mathrm{L}^{\mathrm{O}} \mathrm{T}^{\mathrm{B}} \mathrm{A}^{\mathrm{S}}$, where value of 'Q' and 'R' are
An air filled parallel plate electrostatic actuator is shown in the figure. The area of each capacitor plate is $100 \mu m \times 100 \mu m$. The distance between the plates $d_0 = 1 \mu m$ when both the capacitor charge and spring restoring force are zero as shown in Figure (a). A linear spring of constant $k = 0.01 N/m$ is connected to the movable plate. When charge is supplied to the capacitor using a current source, the top plate moves as shown in Figure (b). The magnitude of minimum charge (Q) required to momentarily close the gap between the plates is ________ $\times 10^{-14}$ C (rounded off to two decimal places).
Note: Assume a full range of motion is possible for the top plate and there is no fringe capacitance. The permittivity of free space is $\epsilon_0 = 8.85 \times 10^{-12}$ F/m and relative permittivity of air ($\epsilon_r$) is 1.
A 60 V DC source with an internal resistance \(R_{int} = 0.5 \, \Omega\) is connected through a switch to a pair of infinitely long rails separated by \(l = 1\) m as shown in the figure. The rails are placed in a constant, uniform magnetic field of flux density \(B = 0.5\) T, directed into the page. A conducting bar placed on these rails is free to move. At the instant of closing the switch, the force induced on the bar is
Match List-I with List-II: List-I