📚 Topic Summary
The Biot-Savart Law allows us to calculate the magnetic field generated by a current-carrying wire. For a circular arc, the magnetic field at the center of the curvature depends on the current, the radius of the arc, and the angle subtended by the arc. The formula simplifies when dealing with arcs of circles, making calculations manageable.
Specifically, the magnetic field $dB$ produced by a small element $dl$ of a current-carrying wire is given by: $dB = \frac{\mu_0}{4\pi} \frac{Idl \times \hat{r}}{r^2}$, where $\mu_0$ is the permeability of free space, $I$ is the current, $dl$ is the length element, $r$ is the distance from the element to the point where the field is being calculated, and $\hat{r}$ is the unit vector in the direction from the current element to the point.
🧠 Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Permeability of Free Space |
A. The angle, in radians, subtended by the arc at the center. |
| 2. Biot-Savart Law |
B. A constant ($\mu_0$) that relates the magnetic field to the current producing it. |
| 3. Magnetic Field |
C. A law describing the magnetic field generated by a constant electric current. |
| 4. Current Element |
D. A region around a magnet or current-carrying wire that exerts a force on other magnets or moving charges. |
| 5. Angle Subtended |
E. A small segment of a current-carrying wire, represented by $Idl$. |
✍️ Part B: Fill in the Blanks
The Biot-Savart Law is crucial for calculating the __________ field produced by a current-carrying __________. For a circular __________, the magnetic field at the center is directly proportional to the __________ and inversely proportional to the __________. The __________ of free space is a constant used in the calculation.
🤔 Part C: Critical Thinking
How does the magnetic field change if the radius of the circular arc is doubled, while the current and angle subtended remain constant? Explain your reasoning.