Doubt from electrostatics

In illustration 1.1 i didn't understand why they have written final value of Q as negative when value of Q was found to be positive

Using calculation they only found out the magnitude of Q
For sign they simply analysed that if the new charge is positive then system can’t be in equilibrium, it must be a -ve charge .
To get magnitudes as well as sign of Q through calculations you’ll have to alter the equation a little bit

There in equation for equilibrium of 1 they have equated force due to Q= force due to q , i.e they have assumed sign of Q is opposite to that of q . And hence only magnitude came up

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Question 2 first part and question 3 second part

2i a)

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2i b)

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3 ii)

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In illustration 1.21 i have a doubt in final result in value of Q 4k should be in denominator not numerator

Yes , it should be in denominator

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Question no 3 part 3 ,Question no 5 and 7

Please help anyone

Q7)

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  1. Simply the field due to only one point charge at centre.

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But answer to this is zero given in answer key

@abhijit_2020 there i missed one thing
For semi infinite wire - it’s kλ/r towards left also , i wrote vertical component as kλ/r but horizontal component kλ/r will also be there

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Question no 2

One more doubt in question 1 that i am uploading i am getting sin@tan@ both in expression of lamda in final answer but one teacher told sin@ will not be there who is correct

But answer is

the tick marked one