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Question

The linear charge density of a thin rod of length L is given by λ=(λ0x) Coulomb/m,[0x<L2] and λ=(λ0x2) Coulomb/m,[L2xL]. Find the total charge on rod. (x is measure from one end of the rod).

A
λ0L224(3+7L)
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B
λ0L38(1+5L)
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C
λ0L24(1+7L)
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D
λ0L2(1+7L)
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Solution

The correct option is A λ0L224(3+7L)
Total charge on AB can be calculated by,

qAB=x=L2x=0λdx

From the data given in the question,

qAB=x=L2x=0λ0x dx

qAB=λ0[x22]x=L2x=0

qAB=λ0L28

Similarly , total charge on BC can be calculated by,

qBC=x=Lx=L2λdx

qBC=x=Lx=L2λ0x2 dx

qBC=λ03[x3]x=Lx=L2

qBC=λ03[L3L38]

qBC=7λ0L324

Total charge on AC,

q=qAB+qBC

q=λ0L28+7λ0L324

q=λ0L224(3+7L)

Hence, option (a) is the correct answer.

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