Solved Problem on Magnetic Field

At a point in São Paulo (a Brazilian city), the earth's magnetic field vector has magnitude \( B_{\small E}=8 \pi \times 10^{–6} \mathrm T \). At this point, a solenoid is placed so that its axis is parallel to the Earth's magnetic field \( {\vec B}_{\small E} \). The length of the solenoid is 0.25 m and it has 500 turns. Calculate the magnitude of its current so that the magnetic field inside it is zero. Vacuum Magnetic Permeability \( \mu_0=4\pi \times 10^{-7}\;\mathrm{\frac{T.m}{A}} \).

Problem data:

Solenoid length: ℓ = 0.25 m;
Number of solenoid turns: N = 500 turns;
Magnetic field at the location: \( B_{\small E}=8 \pi \times 10^{–6} \mathrm T \).
Vacuum Magnetic Permeability: \( \mu_0=4\pi \times 10^{-7}\;\mathrm{\frac{T.m}{A}} \).

Problem diagram:

We assume the direction of the Earth's magnetic field, \( {\vec B}_{\small E} \), is positive (Figure 1).

Figure 1

Solution

The resultant vector of the magnetic field is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {\vec B={\vec B}_{\small S}+{\vec B}_{\small E}} \end{gather} \]

The magnitude for the resultant of the magnetic field to be zero, we have the condition

\[ \begin{gather} B_{\small E}-B_{\small S}=0 \tag{I} \end{gather} \]

The magnitude of the magnetic field of a solenoid is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {B_S=\mu_0\frac{N}{\ell}i} \tag{II} \end{gather} \]

substituting expression (II) for the solenoid magnetic field in equation (I)

\[ \begin{gather} B_{\small E}-\mu_{0}\frac{N}{\ell}i=0\\[5pt] i=\frac{B_{\small E} \ell}{\mu_{0} N}\\[5pt] i=\frac{\left(8\cancel{\pi}\times10^{-6}\;\mathrm{\cancel{T}}\right)\left(0,25\;\mathrm{\cancel{m}}\right)}{\left(4\pi\times 10^{-7}\;\mathrm{\frac{\cancel{T}.\cancel{m}}{A}}\right)\left(500\;\mathrm{turns}\right)} \end{gather} \]

\[ \begin{gather} \bbox[#FFCCCC,10px] {i=0,01\;\mathrm A=10\;\mathrm{mA}} \end{gather} \]

Note: The number of turns, 500, is a dimensionless quantity. "Turns" is not a physical quantity, which is why it does not appear in the final unit of the problem.

Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .