Solved Problem on Magnetic Field

In the figure, the battery maintains a solenoid current with resistance R = 9 Ω. The solenoid has ten turns per centimeter. Assume \( \mu_{0}=4\pi\times 10^{-7}\;\frac{\mathrm{T.m}}{\mathrm{A}} \).
a) Determine the magnitude of the magnetic field vector inside the solenoid;
b) Is the X end of the solenoid a North pole or a South pole?

Problem data:

Solenoid resistance: R = 9 Ω;
Linear density of turns: \( \frac{N}{\ell}=10\;\frac{\text{spires}}{\mathrm{cm}} \);
Battery: E = 100 V;
Vacuum permeability: \( \mu_{0}=4\pi\times 10^{-7}\;\frac{\mathrm{T.m}}{\mathrm{A}} \).

Problem diagram:

We assume the conventional direction of electric current, from positive to negative (Figure 1).

Figure 1

Solution

First, we convert the linear density of turns given in turns per centimeter (turns/cm) to turns per meter used in the International System of Units (SI).

\[ \begin{gather} \frac{N}{\ell}=10\;\frac{\text{spires}}{\cancel{\mathrm{cm}}}\times\frac{100\;\cancel{\mathrm{cm}}}{1\;\mathrm{m}}=1000\;\frac{\text{espiras}}{\mathrm{m}} \end{gather} \]

a) The magnetic field of a solenoid is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {B=\mu_{0}\frac{N}{\ell}i} \tag{I} \end{gather} \]

In the circuit, the resistance of the solenoid and the external resistance are in series, and the equivalent resistance of a series circuit is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {r_{eq}=\sum_{k=1}^{n}r_{k}} \end{gather} \]

for k = 2 reisistors

\[ \begin{gather} r_{eq}=r+R\\[5pt] r_{eq}=1+9\\[5pt] r_{eq}=10\;\Omega \end{gather} \]

The Ohm's Law is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {U=ri} \end{gather} \]

\[ \begin{gather} E=r_{eq}i\\[5pt] i=\frac{E}{r_{eq}}\\[5pt] i=\frac{100}{10}\\[5pt] i=10\;\mathrm{A} \end{gather} \]

Substituting the current calculated above, the problem data, and using π = 3.14 in equation (I), we find the magnitude of the magnetic field

\[ \begin{gather} B=4\times 3.14\times 10^{-7}\times 1000\times 10 \end{gather} \]

\[ \begin{gather} \bbox[#FFCCCC,10px] {B=1.2\times 10^{-2}\;\mathrm{T}} \end{gather} \]

b) To determine whether the X endpoint is a North or South pole, we use the Right Hand Rule. Placing the fingers in the direction of the current, the thumb will indicate the direction of the Magnetic Field lines from South to North (Figure 2).

Figure 2

The X end is a South Pole (Figure 3).

Figure 3

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