A thin wire with positive charge Q evenly spread along its...

A thin wire with positive charge $Q$ evenly spread along its length is shaped into a semicircle of radius $R$. (a). What is the direction of the electric field at the center of curvature of the semicircle? Explain. (b) Is the magnitude of the field at the center less than, equal to, or greater than $k Q / R^{2}$ ? Explain.

A thin wire with positive charge $Q$ evenly spread along its length is shaped into a semicircle of radius $R$.
(a). What is the direction of the electric field at the center of curvature of the semicircle? Explain. (b) Is the magnitude of the field at the center less than, equal to, or greater than $k Q / R^{2}$ ? Explain.

Key Concepts

Comparison with a Point Charge Field

This concept involves comparing the result of the distributed charge’s electric field with that of a point charge. By analyzing how the field due to the semicircular wire relates to the field of a point charge (kQ/R^2), one can assess whether the field at the center is less than, equal to, or greater than the field that would be produced by a point charge with the same total charge at the same distance.

Symmetry in Charge Distributions

Symmetry plays a crucial role in simplifying the calculation of the electric field. In a semicircular arrangement, symmetry arguments help predict the direction of the net field by showing how certain components from opposite segments cancel each other, leaving only the components that add constructively.

Integration of Electric Field Contributions

The integration process involves summing the vector contributions of the electric field from all differential charge elements along the semicircle. This method is used to calculate both the magnitude and the direction of the resultant electric field by taking into account the geometry and charge distribution.

Electric Field from Continuous Charge Distributions

This concept involves calculating the net electric field by integrating the contributions from infinitesimally small charge elements along a continuous charge distribution. In the case of the semicircular wire, Coulomb’s law is applied to each small element, and the vector sum of these contributions gives the net electric field at a point.

Coulomb’s Law

Coulomb’s law is fundamental in determining the electric field created by a point charge. For continuous charge distributions, this law is used differentially by considering small charge segments (dQ) whose electric field contributions are summed up through integration.

Transcript

00:01 Hi, in the given problem there is a wire carrying a charge capital q.

00:09 This wire is bent in the form of a semicircle like this.

00:21 Then this is the diameter of this semicircle.

00:27 Here this will be the center.

00:28 This charge q, capital q, which has been spread over this wire uniformly here.

00:42 It is like this.

00:44 A charge q has been spread over the wire, which has been bent in the form of a semicircle.

01:20 In the first part of the problem, we have to find the direction of net electric field at the center of this semicircle, considering the radius to be r.

01:34 So, first of all, if we consider an elemental charge to be present over here, exactly opposite to this center at this circumference, then due to this elemental charge, dq, the direction of electric field will be away and along the line joining this elemental charge with the observation point.

01:59 This is the electric field de.

02:03 Now if we consider another elemental charge dq here, then the direction of electric field due to this elemental charge will be again away from this charge and along the line joining this charge with the observation point.

02:22 So here this electric field will be if we consider this angle to be theta then this electric field will be having two components.

02:29 One of this is d.

02:31 E...

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