Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

Determine the electric dipole moment of the system of the three charges, placed on the vertices of an equilateral triangle, as shown in the figure :

A

$$2q\ell \widehat j$$

B

$$\left( {q\ell } \right){{\widehat i + \widehat j} \over {\sqrt 2 }}$$

C

$$\sqrt 3 \,q\ell {{\widehat j - \widehat i} \over {\sqrt 2 }}$$

D

$$ - \sqrt 3 \,q\ell \widehat j$$

$$\left| {{P_1}} \right| = $$ q(d)

$$\left| {{P_2}} \right| = $$ qd

|Resultant| $$=$$ 2 P cos30

2 qd$$\left( {{{\sqrt 3 } \over 2}} \right)$$ = $$\sqrt 3 $$ qd

2

MCQ (Single Correct Answer)

Three charges Q, + q and + q are placed at the vertices of a right-angle isosceles triangle as shown below. The net electrostatic energy of the configuration is zero, if the value of Q is :

A

$${{ - q} \over {1 + \sqrt 2 }}$$

B

+ q

C

$$-$$ 2q

D

$${{ - \sqrt 2 q} \over {\sqrt 2 + 1}}$$

U = K$$\left[ {{{{q^2}} \over a} + {{Qq} \over a} + {{Qq} \over {a\sqrt 2 }}} \right] = 0$$

$$ \Rightarrow $$ q = $$-$$ Q$$\left[ {1 + {1 \over {\sqrt 2 }}} \right]$$

$$ \Rightarrow $$ Q = $${{ - q\sqrt 2 } \over {\sqrt 2 + 1}}$$

3

MCQ (Single Correct Answer)

Charges â€“q and +q located at A and B, respectively, constitude an electric dipole. Distance AB = 2a, O is the mid point of the dipole and OP is perpendicular to AB. A charge Q is placed at P where OP = y and y >> 2a. The charge Q experiences an electrostatic force F. If Q is now moved along the equatorial line to P' such that OP' = $$\left( {{y \over 3}} \right)$$, the force on Q will be close to - $$\left( {{y \over 3} > > 2a} \right)$$

A

9F

B

3F

C

F/3

D

27F

Electric field of equitorial plane of dipole

$$ = - {{K\overrightarrow P } \over {{r^3}}}$$

$$ \therefore $$ At P, F $$ = - {{K\overrightarrow P } \over {{r^3}}}$$Q.

At P^{1} , F^{1} $$ = - {{K\overrightarrow P Q} \over {{{\left( {r/3} \right)}^3}}} = 27F.$$

$$ = - {{K\overrightarrow P } \over {{r^3}}}$$

$$ \therefore $$ At P, F $$ = - {{K\overrightarrow P } \over {{r^3}}}$$Q.

At P

4

MCQ (Single Correct Answer)

Two electric dipoles, A, B with respective dipole moments $${\overrightarrow d _A} = - 4qai$$ and $${\overrightarrow d _B} = - 2qai$$ are placed on the x-axis with a separation R, as shown in the figure. The distance from A at which both of them produce the same potential is -

A

$${{\sqrt 2 R} \over {\sqrt 2 + 1}}$$

B

$${R \over {\sqrt 2 + 1}}$$

C

$${{\sqrt 2 R} \over {\sqrt 2 - 1}}$$

D

$${R \over {\sqrt 2 - 1}}$$

V $$ = {{4qa} \over {\left( {R + x} \right)}} = {{2qa} \over {\left( {{x^2}} \right)}}$$

$$\sqrt 2 x = R + x$$

$$x = {R \over {\sqrt 2 - 1}}$$

dist $$ = {R \over {\sqrt 2 - 1}} + R = {{\sqrt 2 R} \over {\sqrt 2 - 1}}$$

$$\sqrt 2 x = R + x$$

$$x = {R \over {\sqrt 2 - 1}}$$

dist $$ = {R \over {\sqrt 2 - 1}} + R = {{\sqrt 2 R} \over {\sqrt 2 - 1}}$$

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