Three long parallel wires are at the corners of a square of side 5.0 cm and each carry a current in the same direction of 99.0 A. The wires are shown as black dots in the figure and are perpendicular to the page or screen. What is the magnitude of the magnetic field at the corner where there is no wire?
Answer (1)
The magnitude of the net magnetic field at the empty corner is 0.396 mT.[tex]________________________________[/tex]To determine the magnitude of the magnetic field at the empty corner of the square, we will use Biot-Savart’s Law or Ampère’s Law, particularly the equation for the magnetic field due to a long, straight current-carrying wire:[tex]\[B = \frac{\mu_0 I}{2 \pi r}\][/tex]where: [tex]\( B \)[/tex] is the magnetic field at distance [tex]\( r \)[/tex], [tex]\( \mu_0 = 4\pi \times 10^{-7} \)[/tex] T·m/A (permeability of free space), [tex]\( I = 99.0 \)[/tex] A (current in each wire), [tex]\( r = 5.0 \)[/tex] cm [tex]\( = 0.050 \)[/tex] m (distance from the wire to the corner). Step 1: Find the Magnetic Field from One Wire For each wire, the magnetic field at the empty corner is:[tex]\[B = \frac{(4\pi \times 10^{-7}) (99.0)}{2\pi (0.050)}\][/tex][tex]\[B = \frac{(4 \times 10^{-7} \times 99)}{(2 \times 0.050)}\][/tex][tex]\[B = \frac{3.96 \times 10^{-5}}{0.1}\][/tex][tex]\[B = 3.96 \times 10^{-4} \text{ T} = 0.396 \text{ mT}\][/tex]Each wire contributes a field of 0.396 mT at the empty corner.Step 2: Find the Net Magnetic Field Since the three wires are at the corners of a square, their magnetic fields form a symmetric pattern at the empty corner. Using the right-hand rule, the fields add up as vectors. By symmetry, the net field is found using vector addition at 120° angles. Using the vector sum formula:[tex]\[B_{\text{net}} = B \sqrt{2 + 2\cos 120^\circ}\][/tex]Since [tex]\(\cos 120^\circ = -\frac{1}{2}\)[/tex], we get:[tex]\[B_{\text{net}} = B \sqrt{2 - 1}\][/tex][tex]\[B_{\text{net}} = B \times \sqrt{1}\][/tex][tex]\[B_{\text{net}} = 0.396 \text{ mT}\][/tex]Thus, the magnitude of the net magnetic field at the empty corner is 0.396 mT.
