# A circular ring of radius r made of a non-conducting material is placed with its axis parallel to a uniform electric field. The ring is rotated about a diameter through 180°. Does the flux of electric field change? If yes, does it decrease or increase?

No, the flux will remain the same.

We know, Flux = Where

E = electric field vector, ds = small area element.

Now, ds = n.da,

where n is the unit normal in the direction of the area, and da is a small area element.

By the law of dot product in vector calculus,

E.ds = |E||ds|cosθ

Where

θ is the angle between the normal to the area and the electric field.

|E| = magnitude of electric field vector,

|ds| = magnitude of small area element.

In the first case, since the axis of the ring is parallel to the electric field, the angle θ between the normal to the surface is 0°, since the normal is parallel to the axis.

When θ=180o

Φ=|E||ds| cos(180o) =|E||ds| (-1) =-|E||ds|

Therefore, the flux is just |E||ds|. When it is rotated about its diameter by 180°, the normal just rotates by 180°, and thus the angle changes to 180°. The flux becomes negative, but its value remains the same.

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