The field at the centre O of a semi circular wire of radius r(see fig.) if the straight wires are of infinite length is:-
The net field at point O is = field due to the straight wires + field due to the arc
first we'll try to understand the straight line case:
We'll divide it into two separate discussions
In the above case the formula we use for finding out the magnetic field at a particular point at space is given by:
Here d is the perpendicular distance from the mid point of wire to the point concerned.
But, the situation given in the question is a modification of the above situation.
In the above diagram,
The arrow on the black wire shows the direction of current.
The red arrow from the black wire heads towards the point at which we have to find the magnetic field.
Therefore the red line is r.
And the point is O. I forgot to show it in the diagram.
So we'll solve the straight lines part first:
Using Fleming's rule or right hand palm rule, we know that for both the wires, the magnetic field is in the same direction, so we find the field due to one and multiply it by 2.
The above Field is towards the observer.
The magnetic field due to a current carrying circular loop is given by:
So the formula to find the magnetic field due to the arc:
Here alpha radian is the angle subtended by the arc.
Here alpha = pi.
the Magnetic field due to the arc is:
Now subtract the magnetic field due to straight lines from the magnetic field due to arc.