Current-carrying wires generate a magnetic field around them.
Used Biot-Savart Law to determine magnetic field strength and direction around a straight wire.
Discussed infinite and semi-infinite wires and their corresponding formulas.
Solved various application problems related to the above concepts.
Today's Lecture Focus
Analyzing magnetic fields created by circular wires, semicircles, and quadrants.
Calculating fields specifically at the center of these shapes.
Section 1: Magnetic Field Due to Circular Current-Carrying Loop
Understanding the Circular Loop
A loop carrying current generates magnetic field at its center.
Biot-Savart Law is applied:
$$ B = \frac{\mu_0}{4\pi} \cdot \frac{I \cdot dL \cdot \sin(\theta)}{r^2} $$
At the center, all dL vectors are perpendicular to the radius vector, leading to:
$$ B_{center} = \frac{\mu_0 I}{2 A} $$
Where A is the radius of the circular loop.
Direction of Magnetic Field
Right-hand rule to determine direction:
Thumb points in the direction of current, fingers curl in the direction of the magnetic field.
Field points inward if the current is counterclockwise.
Section 2: Semi-Circular and Quadrant Shapes
Magnetic Field from a Semi-Circular Wire
Formula:
$$ B = \frac{\mu_0 I}{4 A} $$
Direction is determined similarly with right-hand rule.
Magnetic Field from a Quadrant
Field is found to be half of the semi-circle's magnetic field.
Formula:
$$ B = \frac{\mu_0 I}{8 A} $$
Section 3: General Angle Theta Arc
For a circular arc at center:
$$ B = \frac{\mu_0 I \theta}{4 \pi A} $$
Theta should be in radians.
Right-hand rule applies for determining direction again.
Section 4: Changing Current Direction
If the current direction is reversed, the magnetic field direction also reverses.
Section 5: Application Questions
Parallel Wires and Circular Directions
Magnetic fields cancel when both currents are equal and opposite.
Case of points along a straight wire results in zero magnetic field.
Complex Configurations
Magnetic fields in composite wires require vector addition to find net field.
When cases involve resistance in the circuit, calculations may adjust based on setup.
Conclusion
Emphasized thorough practice with questions related to magnetic field setup in circular or complex arrangements to solidify understanding. Next, the focus will be on magnetic fields on the axis of circular current-carrying wires.
Recommended Practice:
Solve related problems and ensure clarity on formulas and application of the right-hand rule throughout different scenarios.
Further Resources:
For comprehensive notes, check the website mentioned in the lecture.