NET 2021 Physics - Vector analysis

 Question:

The volume integral   

 

 

is over a region τ bounded by a surface Ʃ ( an infinitesimal area element being

 

 

where  

 is the outward unit normal. If it changes to

 

 

When the vector 

 is changed to

 

 

Then 

 can be expressed as

 

 Option a) 

 

Option b ) 

b) 

Option c) 

c)  

Option d) 

Solution :

Given

 

 

As per the given problem 

 changes to

Substituting for 

we get 

which  reduces  to

 

.                                                             

As curl gradient of a scalar function is zero

The above equation can be written as

Hence

Let

 

 

then

                                                                                                                 …………… (1)

Now using the vector  identity

The first tem of R.H.S becomes zero as    

 

 

And curl gradient of a scalar function is zero. Therefore

Substituting  the above equation in equation 1.

Using divergence theorem

Substituting for

 and simplifying

 So option c is the answer. 

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