12.3 Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law

• What is the value of the permeability of free space, $\mu_{0}$?


  $4\pi \times 10^{ - 7} \, T \cdot m / A$
• What does $r$ represent in the Biot-Savart Law?


  Distance to the point
• What is the magnetic field calculated for a $0.01 \, m$ wire carrying $5 \, A$ at a distance of $0.1 \, m$?


  $5 \times 10^{ - 6} \, T$
• In the Biot-Savart Law, $\mu_{0}$ is the permeability of free space
• The permeability of free space, $\mu_{0}$, is equal to 4\pi \times 10^{ - 7} \, T \cdot m / A
• What does $\mathbf{dl}$ represent in the Biot-Savart Law?


  Infinitesimal current element
• What does the Biot-Savart Law calculate?

  Magnetic field
• What is the source of the Biot-Savart Law compared to Coulomb's Law?

  Current elements
• The unit vector $\hat{\mathbf{r}}$ in the Biot-Savart Law points from the current element to the point of observation.


  True
• What is the SI unit for the permeability of free space ($\mu_{0}$)?


  $T \cdot m / A$
• What does the Biot-Savart Law calculate?

  Magnetic field
• What is the source of the magnetic field in the Biot-Savart Law?

  Current elements
• What is the magnitude of the magnetic field for a $0.01 \, m$ wire carrying $5 \, A$ at $0.1 \, m$ distance?


  $5 \times 10^{ - 6} \, T$
• Steps to determine the direction of the magnetic field using the right-hand rule for $\mathbf{dl}$ and $\hat{\mathbf{r}}$


  1️⃣ Point thumb along $\mathbf{dl}$

  2️⃣ Curl fingers along $\hat{\mathbf{r}}$

  3️⃣ Magnetic field direction is perpendicular to palm
• The right-hand rule is used to determine the direction of the magnetic field for a circular loop.

  True
• The variable $I$ in the magnetic field formula represents the current in the loop.


  True
• The Biot-Savart Law is expressed as: \mathbf{dB} = \frac{\mu_{0}}{4\pi} \frac{I \mathbf{dl} \times \hat{\mathbf{r}}}{r^{2}}
• Match the variables in the Biot-Savart Law with their meanings:

  \mathbf{dB} ↔️ Infinitesimal magnetic field vector

  \mu_{0} ↔️ Permeability of free space

  I ↔️ Current in the wire

  \mathbf{dl} ↔️ Infinitesimal vector current element
• The Biot-Savart Law depends on the permeability of free space, the current element, the distance, and the unit vector
• Match the features with the correct law:

  Biot-Savart Law ↔️ Magnetic field from a current element

  Coulomb's Law ↔️ Electric field from a point charge
• The Biot-Savart Law describes the magnetic field from a current element, while Coulomb's Law describes the electric field from a point charge.


  True
• The Biot-Savart Law and Coulomb's Law both depend on the distance from the source.


  True
• The distance $r$ in the Biot-Savart Law is measured between the current element and the point where $\mathbf{dB}$ is calculated.


  True
• The term $I$ in the Biot-Savart Law represents the current flowing in the wire.


  True
• For a $0.01 \, m$ wire carrying $5 \, A$ at $0.1 \, m$ distance, what is the magnetic field $\mathbf{dB}$?


  $5 \times 10^{ - 6} \, T$
• The direction of $\mathbf{dl}$ in the Biot-Savart Law is along the wire, pointing in the direction of current flow
• $\mu_{0}$ is the permeability of free space, which has a value of $4\pi \times 10^{ - 7}$ T \cdot m / A
• The Biot-Savart Law uses $\mathbf{dl}$ to represent the infinitesimal current element
• The Biot-Savart Law equation is \mathbf{dB} = \frac{\mu_{0}}{4\pi} \frac{I \mathbf{dl} \times \hat{\mathbf{r}}}{r^{2}}</latex>, while Coulomb's Law equation is $\frac{1}{4\pi \epsilon_{0}} \frac{q \hat{\mathbf{r}}}{r^{2}}$.Coulomb's
• The unit vector $\hat{\mathbf{r}}$ points from the current element to the point of observation.


  True
• What is the magnetic field direction for an infinite straight wire?

  Circular around the wire
• What is the magnetic field at the center of a circular loop with current $I$ and radius $R$?


  $\mathbf{B} =$$\frac{\mu_{0}I}{2R}$
• In the magnetic field formula, \mu_{0}</latex> represents the permeability of free space
• The magnetic field along the axis of a circular loop is given by $\mathbf{B} =$ \frac{\mu_{0}I}{2}\left(\frac{R^{2}}{(R^{2} + z^{2})^{3 / 2}}\right), where $z$ is the distance along the axis
• What is the general formula for the Biot-Savart Law?

  $\mathbf{dB} =$$\frac{\mu_{0}}{4\pi} \frac{I \mathbf{dl} \times \hat{\mathbf{r}}}{r^{2}}$
• What is the first step in calculating the magnetic field at the center of a circular loop using the Biot-Savart Law?

  Divide the loop into elements
• What does the Biot-Savart Law calculate?

  Magnetic field of a current
• The Biot-Savart Law describes magnetic fields
• In the Biot-Savart Law, $\mu_{0}$ represents the permeability of free space
• What type of field does the Biot-Savart Law calculate?

  Magnetic field