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Figure shows a conducting loop ABCDA placed in a uniform magnetic field (strength B) perpendicular to its plane. The part ABC is the (three-fourth) portion of the square of side length l. The part ADC is a circular arc of radius R. The points A and C are connected to a battery which supplies a current I to the circuit. The magnetic force on the loop due to the field B is:

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A. zero
B. BIl
C. 2BIR
D. BIlRI+R

Answer
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Hint: Since, the current through the loop ABCDA splits into to part, one flows through ABC and other through ADC. During the current flow, there will be some force generated. To find the total force generated in the loop ABCDA, the force generated on loops ABC and ADC are calculated. Since, the two currents travel the same distance in the magnetic field, the current gets added. Thus, the total force can be calculated.

Useful formula:
Relation between force, current and magnetic field will be given by,
F=BIl
Where, F is the force on the conductor, B is the magnetic field, I is the current flows through the conductor and l is the length of the conductor.

Complete Step by step solution:

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Assume that,
The current through the loop ABC is I1
The current through the loop ADC is I2
The length where the current flow in loop ABC is l1
The length where the current flow in loop ADC is l2

The force generated on the loop ABC,
FABC=B×I1×l1......................................(1)
Where, FABC is the force generated on loop ABC, I1 is the current flows through the loop ABC and l1 is the length of the loop ABC.

The force generated on the loop ADC,
FADC=B×I2×l2......................................(2)
Where, FADC is the force generated on loop ADC, I2 is the current flows through the loop ADC and l2 is the length of the loop ADC.

Hence, the total force on loop ABCDA is the sum of the force on loop ABC and the force on loop ADC,
F=FABC+FADC..........................................(3)
Substitute the values of (1) and (2) equation (1),
F=(B×I1×l1)+(B×I2×l2)F=B[(I1×l1)+(I2×l2)]

Since, the current travels the same length in the loops ABC and ADC ,
 l=l1=l2
Hence,
F=B[(I1×l)+(I2×l)]F=B×l[I1+I2]
The total current, I=I1+I2
Thus,
F=B×l[I]F=BIl

Hence, the option (B) is correct.

Note: In the loop ABCDA, even the structure of the loops ABC and ADC may vary, but the current flows through the loops in the uniform magnetic field travels the same distance in both the loops. A force will be generated in the current carrying conductor placed in a uniform magnetic field, this is the statement of Faraday’s law of induction. The force generated in the loop ABCDA, is the sum of the force generated in loops ABC and ADC.
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