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Zorluk: ZorMagnetic Force and Electromagnetism

A straight copper rod of length 0.50 m0.50\text{ m} and mass 40 g40\text{ g} carries a steady current and is suspended horizontally in a uniform magnetic field of 0.40 T0.40\text{ T}. The field is directed horizontally at an angle of 30∘30^\circ to the length of the rod. If the upward magnetic force acting on the rod exactly balances its weight, what is the magnitude of the current flowing through the rod? (Take g=10 m/s2g = 10\text{ m/s}^2)

  1. A
    0.25 A0.25\text{ A}
  2. B
    2.0 A2.0\text{ A}
  3. 4.0 A4.0\text{ A}Cevap
  4. D
    40 A40\text{ A}

Cevap

The magnitude of the current required to balance the weight of the rod is 4.0 A4.0\text{ A}.
For the rod to remain suspended in equilibrium, the upward magnetic force FB=BILsin⁡θF_B = BIL \sin \theta must balance the downward gravitational force W=mgW = mg. Substituting m=0.040 kgm = 0.040\text{ kg}, g=10 m/s2g = 10\text{ m/s}^2, B=0.40 TB = 0.40\text{ T}, L=0.50 mL = 0.50\text{ m}, and θ=30∘\theta = 30^\circ yields 0.40=0.40×I×0.50×0.500.40 = 0.40 \times I \times 0.50 \times 0.50, which solves to I=4.0 AI = 4.0\text{ A}.

Adım Adım Çözüm

1
Convert the mass of the rod to kilograms and calculate its weight.
m=40 g=0.040 kgm = 40\text{ g} = 0.040\text{ kg}, so W=mg=0.040 kg×10 m/s2=0.40 NW = mg = 0.040\text{ kg} \times 10\text{ m/s}^2 = 0.40\text{ N}.
Standard SI units must be used to ensure dimensional consistency.
2
Express the magnetic force acting on the conductor in terms of current II.
FB=BILsin⁡θ=(0.40 T)×I×(0.50 m)×sin⁡30∘=0.10I NF_B = B I L \sin \theta = (0.40\text{ T}) \times I \times (0.50\text{ m}) \times \sin 30^\circ = 0.10 I\text{ N}.
The magnetic force on a current-carrying conductor at an angle θ\theta to a magnetic field is given by F=BILsin⁡θF = BIL \sin \theta.
3
Equate the upward magnetic force to the downward weight to find II.
0.10I=0.40  ⟹  I=0.400.10=4.0 A0.10 I = 0.40 \implies I = \frac{0.40}{0.10} = 4.0\text{ A}.
For vertical equilibrium, the net vertical force must equal zero.

Anahtar Kavram

Equilibrium of a current-carrying conductor in a uniform magnetic field (F=BILsin⁡θ=mgF = BIL \sin \theta = mg)
Tahmini Süre:2m 0s
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Konular

Magnetic Force and Electromagnetism

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