Question

Consider the algebraic expression for the electric field of a uniformly charged ring, at a location on the axis of the ring. Q is the charge on the entire ring, and $\mathbf{∆}\mathit{Q}$is the charge on one piece of the ring. $\mathbf{∆}\mathit{\theta }$is the angle subtended by one piece of the ring (or, alternatively, $\mathbf{∆}\mathit{r}$is the arc length of one piece). What is$\mathbf{∆}\mathit{Q}$, expressed in terms of given constants and an integration variable? What are the integration limits?

Short answer

Answer

The value of $\mathbf{∆}\mathit{Q}$is $\frac{\mathbf{Q}}{\mathbf{2}\mathbf{\pi }}\left(∆\theta \right)$.

The integration limits are $\mathbf{0}$to $\mathbf{2}\mathit{\pi }$.

Step-by-step solution

Identification of the given data

The given data is listed below as:

• The ring’s charge is, Q

• The charge of the one single piece of the ring is, $\mathbf{∆}\mathit{Q}$

• The angle created by the ring’s one piece is, $\mathbf{∆}\mathit{\theta }$

• The one piece’s arc length is, $\mathbf{∆}\mathit{r}$

Significance of the electric field

The electric field is independent on the test charge amount and it is described as the property of the system of various charges. The electric field also gives direction and magnitude of a particular electric force.

Determination of the charge ∆Q

For a particular charged ring, the product of the length of the ring’s piece and charge’s linear density gives the charge of one single piece of a ring. As there is uniform distribution of charges, the linear density of the ring is described as the division of the total charge and the ring’s total length.

The length of one single piece of a ring is described as the length of a particular arc that is subtended by a particular angle $\mathbf{∆}\mathit{\theta }$which eventually becomes $\mathit{r}\mathbf{∆}\mathit{\theta }$.

The equation of the charge of the ring is expressed as:

$\mathbf{∆}\mathit{Q}\mathbf{=}\frac{\mathbf{Q}}{\mathbf{L}}\mathbf{(}\mathbf{∆}\mathbf{r}\mathbf{)}$

Here, $\mathit{Q}$is the entire ring’s charge, $\mathit{L}$is the ring’s length and $\mathbf{∆}\mathit{r}$is the length of one ring’s piece.

Substitute $\mathit{r}\mathbf{∆}\mathit{\theta }$for $\mathbf{∆}\mathit{r}$and role="math" localid="1657172632070" $\mathbf{2}\mathit{\pi }\mathit{r}$for $\mathit{L}$in the above equation.

$$\begin{gathered} \mathbf{∆}\mathit{Q}\mathbf{=}\frac{\mathbf{Q}}{\mathbf{2}\mathbf{\pi r}}\mathbf{(}\mathbf{r}\mathbf{∆}\mathbf{\theta }\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{Q}}{\mathbf{2}\mathbf{\pi }}\mathbf{(}\mathbf{∆}\mathbf{\theta }\mathbf{)} \end{gathered}$$

Thus, the value of $\mathbf{∆}\mathit{Q}$is $\frac{\mathbf{Q}}{\mathbf{2}\mathbf{\pi }}\mathbf{(}\mathbf{∆}\mathbf{\theta }\mathbf{)}$.

Determination of the integration limits

The integration variable is the angle $\mathit{\theta }$. Hence, because of summing up all ring’s parts, the integration limits for the angle $\mathit{\theta }$is $\mathbf{0}$to $\mathbf{2}\mathit{\pi }$.

Thus, the integration limits are $\mathbf{0}$to $\mathbf{2}\mathit{\pi }$.