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Section 27.2 Magnetic Field: Wire
Checkpoint 27.2.1 . Cross Product Practice.
Consider the three vectors shown below.
Figure 27.2.2.
(a) Indicate the direction of the cross product
\(\vec{v}_2 \times \vec{v}_1\text{.}\)
Hint .
Recall the
Right Hand Rule and remember that the order of the vectors matters!
Answer .
(b) Indicate the direction of the cross product
\(\vec{v}_3 \times \vec{v}_1\text{.}\)
Answer .
Down and to the left, roughly in the direction of
\(\vec{v}_2\text{.}\)
As with electric fields, magnetic fields exist everywhere in space. Many things can create magnetic fields, including electric currents.
Definition 27.2.3 . Magnetic Field Created by a Current.
The magnetic field created by a long, straight wire carrying constant current \(\vec{I}\) a distance \(s\) away is given by
\begin{equation*}
\vec{B} = \frac{\mu_0}{2\pi}\frac{\vec{I} \times \hat{s}}{s}
\end{equation*}
\(\mu_0 = 4\pi \times 10^{-7} \mathrm{~\frac{Tm}{A}}\) is a fundamental constant of the universe known as the permeability of free space.
The direction of
\(\vec{I}\) is the direction of positive charge flow.
\(\hat{s} \) is a
Unit Vector that points radially away from the wire, perpendicular to the current. Note that this
also depends on where you are in space!
Figure 27.2.4. A test point near a long, current-carrying wire. The direction of the magnetic field at the marked point is out of the screen .
Subsubsection Activities
Activity 27.2.1 . Magnetic Field of a Wire.
The wire shown below carries current
\(I\) out of the screen (directly at you!). The gray circle represents a distance
\(R\) away from the wire.
Figure 27.2.5. A wire surrounded by four marked points.
(a) Explanation.
Sketch a vector at each marked point to represent the magnetic field. Explain how you determined each vector.
Answer .
Figure 27.2.6. The magnetic field around a long straight wire.
(b) Calculation.
Calculate the magnitude of the magnetic field at each of the marked points.
Answer .
\begin{equation*}
B = \frac{\mu_0 I}{2\pi R}
\end{equation*}
(c) Sensemaking: Covariational Reasoning.
How does the strength of the magnetic field change as you get farther away from the wire? Explain your reasoning.
Answer .
Since
\(R\) is in the denominator, as
\(s\) increases the magnetic field strength will decrease!
(d) Represent: Vector Field Map.
Use your answers to the above questions to sketch a magnetic field vector map for the region around the wire. Your map should highlight all the major features of the magnetic field you have identified so far.
Answer .
Figure 27.2.7. A magnetic field vector map for a long straight wire. The field points in a counter-clockwise direction and gets smaller as you get farther from the wire.
Figure 27.2.8. Some magnetic field lines of a long straight wire. The direction of \(\vec{B}\) can be found with a form of the right-hand rule.
References References
[1]
“Magnetic Field Wire”
OpenStax , https://openstax.org/books/university-physics-volume-2/pages/12-2-magnetic-field-due-to-a-thin-straight-wire
