A circular arc of charge having radius a is centered on the +x-axis and subtends

Question

A circular arc of charge having radius a is centered on the +x-axis and subtends an angle of theta both above and below the x-axis. If the arc has a linear density of lambda = lambda(subscript)0 cos theta, what is the electric field at the center of the arc? Give both magnitude and direciton (assume lambda(subscript)0 is positive). You may need the identity cos^2 theta = (1 + cos (2 times theta) / 2).

2. A circular are of charge having radius a is centered on the X-axis and subtends an angle of a both above and below the X-axis. If the arc has a linear charge density of a = A, cost, what is the electric field at the center of the are? Give both magnitude and direction (assume A is positive). You may need the identity cost = (1+cos 29)/2.

Explanation / Answer

consider a small element of angular width d(phi) at angle phi
length of this arc , dl = a*d(phi) where a is radius of the arc
charge on this element, dq = a*d(phi)*lambdao*cos(phi)
electric field at the centre, E = kdq/a^2
now only cos component of this field will add as sin component will be cancelled out ( from symmetry)
so dE = kdqcos(phi)/a^2 = k*a*d(phi)*lambdao*cos^2(phi)/a^2
integrating from phi = -theta to +theta
E = integrate(k*a*d(phi)*lambdao*cos^2(phi)/a^2) = integrate[ k*lambdao*(1 + cos(2phi))d(phi)/2a ] = k*lambdao[2*theta + sin(2*theta)]/2a

direction, along -ve x axis

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.