Why Does the Solution Manual Use \( \frac{Q}{\pi R} \cdot d\theta \cdot R \)?

[mrtubby]

Homework Statement

Charged Wire laid out in a semicircle, test charge at the center of the "Circle"
what is the charge of dQ. This isn't my whole question but is the bit of it that I can't understand the actual question asks you to go the whole nine yards with calculating the force on the test charge

The Attempt at a Solution

Charge density: $$\frac{total charge}{length of tiny wire piece}=\frac{Q}{\frac{2\pi R}{2}} = \frac{Q}{\pi R}$$

so charge of wire chunk = $\frac{Q}{\pi R} * d\theta$

when my solution manual uses the charge of the wire chunk it uses $\frac{Q}{\pi R} * d\theta * R$

I can't figure out why, any hints?

 

[Doc Al]

The charge density is charge per unit length. The length of that wire segment is R*dθ, not just dθ.