A dipole of Dipole moment `vec(p)=(p_(0))/(sqrt(2))(hat(i)+hat(j))`. Is placed at origin. Now a uniform external electrical filed at magnitude `E_(0)` is applied along direction of dipole. Two points A and B are lying on a equipotential surface of radius R centered at origin. A is along axial position of dipole and B is along equatorial position. There correct option are :

Options

A. `R=(P_0/(4piepsilon_0E_0))^(1//3)`
B. The magnitude of total electric field on any two points of the circle will be same
C. Total electric field at point A is `vecE_A=sqrt2E_0 (hati+hatj)`
D. Total electric field at point B is `vecE_B=0`

Answer

A, D

Explanation

`vecP=P_0/sqrt2(hati+hatj)`
Hence net dipole moment is `P_0` and dipole is inclined at an angle `45^@` with the positive x-axis. So point A lies on the axis of dipole and point B lies on the perpendicular bisector of the dipole. External uniform electric field `E_0` is also inclined at an angle `45^@` with the positive x-axis.
Electric field due to dipole at point A: `E_A=(2P_0)/(4piepsilon_0R^3)`
Electric field due to dipole at point B: `E_B=P_0/(4piepsilon_0R^3)`
Option (a)- At point B `E_0` and `E_A` are along the tangent to circle along opposite directions as shown in figure. Since circle is equipotential hence electric field along the tangent must be zero at point B.
`rArr E_0=E_B rArr E_0= P_0/(4piepsilon_0R^3)`
`rArr R=(P_0/(4piepsilon_0E_0))^(1//3)` , Hence option (a) is correct .
Option (b)- We know that electric field due to electric dipole is different at different points of the circle and external field is uniform hence net electric field will be different at different points of the circle. Hence option (b) is wrong.
Option (c) - Electric field due to electric dipole at point A is `E_A=(2P_0)/(4piepsilon_0R^3)=2E_0` and external electric field `E_0` is also along the same direction. hence net electric field at point A is
`E_"net"^A=3E_0 ((hati+hatj)/sqrt2)` , Hence option (c ) is wrong.
Option (d) - We already know that net eletric field at point B is zero hence option (d) is correct.