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Sagot :
To determine the magnitude of the magnetic force on each wire and rank them accordingly, we need to use the relationship for the magnetic force acting on a current-carrying wire:
[tex]\[ F = I \cdot L \cdot B \cdot \sin(\theta) \][/tex]
Here, [tex]\( I \)[/tex] is the current, [tex]\( L \)[/tex] is the length of the wire in the magnetic field, [tex]\( B \)[/tex] is the magnetic field strength, and [tex]\( \theta \)[/tex] is the angle between the direction of the current and the magnetic field direction (which is due east in this problem). For simplicity in comparison, we can assume that [tex]\( L \)[/tex] and [tex]\( B \)[/tex] are constants. Therefore, the magnetic force [tex]\( F \)[/tex] will be proportional to [tex]\( I \cdot \sin(\theta) \)[/tex].
Let’s analyze each wire:
1. Wire A:
- Current, [tex]\( I_A = 15 \)[/tex] A
- Angle [tex]\( \theta_A = 45^{\circ} \)[/tex] south of east
- The sine of the angle, [tex]\( \sin(45^{\circ}) = \sqrt{2}/2 \approx 0.707 \)[/tex]
- Proportional force for wire A:
[tex]\[ F_A \propto 15 \cdot \sin(45^{\circ}) = 15 \cdot 0.707 = 10.61 \][/tex]
2. Wire B:
- Current, [tex]\( I_B = 18 \)[/tex] A
- Angle [tex]\( \theta_B = 180^{\circ} \)[/tex] (due west)
- The sine of the angle, [tex]\( \sin(180^{\circ}) = 0 \)[/tex]
- Proportional force for wire B:
[tex]\[ F_B \propto 18 \cdot \sin(180^{\circ}) = 18 \cdot 0 = 0 \][/tex]
3. Wire C:
- Current, [tex]\( I_C = 10 \)[/tex] A
- Angle [tex]\( \theta_C = 60^{\circ} \)[/tex] south of east
- The sine of the angle, [tex]\( \sin(60^{\circ}) = \sqrt{3}/2 \approx 0.866 \)[/tex]
- Proportional force for wire C:
[tex]\[ F_C \propto 10 \cdot \sin(60^{\circ}) = 10 \cdot 0.866 = 8.66 \][/tex]
Now, we compare the magnitudes of the proportional forces:
- [tex]\( F_B \approx 0 \)[/tex]
- [tex]\( F_C \approx 8.66 \)[/tex]
- [tex]\( F_A \approx 10.61 \)[/tex]
Ranking the wires in terms of the magnitude of the magnetic force, from smallest to largest:
[tex]\[ F_B < F_C < F_A \][/tex]
Thus, the correct order is:
[tex]\[ B < C < A \][/tex]
And the corresponding option is:
[tex]\[ \boxed{B < C < A} \][/tex]
[tex]\[ F = I \cdot L \cdot B \cdot \sin(\theta) \][/tex]
Here, [tex]\( I \)[/tex] is the current, [tex]\( L \)[/tex] is the length of the wire in the magnetic field, [tex]\( B \)[/tex] is the magnetic field strength, and [tex]\( \theta \)[/tex] is the angle between the direction of the current and the magnetic field direction (which is due east in this problem). For simplicity in comparison, we can assume that [tex]\( L \)[/tex] and [tex]\( B \)[/tex] are constants. Therefore, the magnetic force [tex]\( F \)[/tex] will be proportional to [tex]\( I \cdot \sin(\theta) \)[/tex].
Let’s analyze each wire:
1. Wire A:
- Current, [tex]\( I_A = 15 \)[/tex] A
- Angle [tex]\( \theta_A = 45^{\circ} \)[/tex] south of east
- The sine of the angle, [tex]\( \sin(45^{\circ}) = \sqrt{2}/2 \approx 0.707 \)[/tex]
- Proportional force for wire A:
[tex]\[ F_A \propto 15 \cdot \sin(45^{\circ}) = 15 \cdot 0.707 = 10.61 \][/tex]
2. Wire B:
- Current, [tex]\( I_B = 18 \)[/tex] A
- Angle [tex]\( \theta_B = 180^{\circ} \)[/tex] (due west)
- The sine of the angle, [tex]\( \sin(180^{\circ}) = 0 \)[/tex]
- Proportional force for wire B:
[tex]\[ F_B \propto 18 \cdot \sin(180^{\circ}) = 18 \cdot 0 = 0 \][/tex]
3. Wire C:
- Current, [tex]\( I_C = 10 \)[/tex] A
- Angle [tex]\( \theta_C = 60^{\circ} \)[/tex] south of east
- The sine of the angle, [tex]\( \sin(60^{\circ}) = \sqrt{3}/2 \approx 0.866 \)[/tex]
- Proportional force for wire C:
[tex]\[ F_C \propto 10 \cdot \sin(60^{\circ}) = 10 \cdot 0.866 = 8.66 \][/tex]
Now, we compare the magnitudes of the proportional forces:
- [tex]\( F_B \approx 0 \)[/tex]
- [tex]\( F_C \approx 8.66 \)[/tex]
- [tex]\( F_A \approx 10.61 \)[/tex]
Ranking the wires in terms of the magnitude of the magnetic force, from smallest to largest:
[tex]\[ F_B < F_C < F_A \][/tex]
Thus, the correct order is:
[tex]\[ B < C < A \][/tex]
And the corresponding option is:
[tex]\[ \boxed{B < C < A} \][/tex]
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