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An infinitely long solenoid of radius r and number of turns per unit length n carries a steady current I. The ratio of the magnetic fields at a point on the axis of the solenoid to a point r/2 from the axis is _______.
Correct answer is '1 to 1'. Can you explain this answer?
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An infinitely long solenoid of radius r and number of turns per unit l...
Explanation:
The ratio of magnetic fields at a point on the axis of the solenoid to a point r/2 from the axis is to be determined.
Magnetic field at a point on the axis of the solenoid:
The magnetic field at a point on the axis of the solenoid can be determined using Ampere's law. Consider a circular path of radius r, which is concentric with the solenoid and lies on the axis of the solenoid. The current enclosed by this circular path is given by Ienc = nπr2I. The magnetic field at any point on this circular path is given by B = μ0Ienc/2r, where μ0 is the permeability of free space. Substituting the value of Ienc, we get B = μ0nI.
Magnetic field at a point r/2 from the axis:
The magnetic field at a point r/2 from the axis can be determined using Biot-Savart law. Consider a circular loop of radius r/2, which is concentric with the solenoid and lies on a plane perpendicular to the axis of the solenoid. The current passing through this circular loop is given by Iloop = nπrI. The magnetic field at the center of this circular loop is given by B = μ0Iloop/2r, where μ0 is the permeability of free space. Substituting the value of Iloop, we get B = μ0nI.
Ratio of magnetic fields:
The ratio of magnetic fields at a point on the axis of the solenoid to a point r/2 from the axis is given by Baxis/Br/2 = (μ0nI)/(μ0nI) = 1.
Therefore, the ratio of the magnetic fields at a point on the axis of the solenoid to a point r/2 from the axis is 1 to 1.
The ratio of magnetic fields at a point on the axis of the solenoid to a point r/2 from the axis is to be determined.
Magnetic field at a point on the axis of the solenoid:
The magnetic field at a point on the axis of the solenoid can be determined using Ampere's law. Consider a circular path of radius r, which is concentric with the solenoid and lies on the axis of the solenoid. The current enclosed by this circular path is given by Ienc = nπr2I. The magnetic field at any point on this circular path is given by B = μ0Ienc/2r, where μ0 is the permeability of free space. Substituting the value of Ienc, we get B = μ0nI.
Magnetic field at a point r/2 from the axis:
The magnetic field at a point r/2 from the axis can be determined using Biot-Savart law. Consider a circular loop of radius r/2, which is concentric with the solenoid and lies on a plane perpendicular to the axis of the solenoid. The current passing through this circular loop is given by Iloop = nπrI. The magnetic field at the center of this circular loop is given by B = μ0Iloop/2r, where μ0 is the permeability of free space. Substituting the value of Iloop, we get B = μ0nI.
Ratio of magnetic fields:
The ratio of magnetic fields at a point on the axis of the solenoid to a point r/2 from the axis is given by Baxis/Br/2 = (μ0nI)/(μ0nI) = 1.
Therefore, the ratio of the magnetic fields at a point on the axis of the solenoid to a point r/2 from the axis is 1 to 1.
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An infinitely long solenoid of radius r and number of turns per unit length n carries a steady current I. The ratio of the magnetic fields at a point on the axis of the solenoid to a point r/2 from the axis is _______.Correct answer is '1 to 1'. Can you explain this answer?
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