If a segment of wire has a net charge (uniformly distributed), and then its length is increased by 50%,by what factor has its charge density changed?

If the original charge was 10.0 nC, how much charge must then be added to restore the wire’s original charge
density?

3/2 the length

same charge
so 2/3 the density

(3/2) * 10 = 30/2 = 15
so increase by 5 nC

How is it 3/2 the length?

To determine the factor by which the charge density has changed, we need to understand how charge density is defined. Charge density is the amount of charge per unit length. Mathematically, it can be expressed as:

Charge density = Charge / Length

Now let's consider the given scenario. The segment of wire initially has a charge of 10.0 nC. If we increase its length by 50%, the new length becomes 1.5 times the original length.

1. To calculate the new charge density, we need to determine the new charge and the new length.

New length = 1.5 * original length (since the length increased by 50%)
New charge = original charge (since no new charge is added)

2. Now we can calculate the new charge density using the formula mentioned earlier.

New charge density = New charge / New length

3. To find out the factor by which the charge density has changed, we need to compare the new charge density with the original charge density.

Charge density change factor = New charge density / Original charge density

4. Finally, to determine the amount of charge that must be added to restore the wire's original charge density, we need to find the difference between the new charge and the original charge.

Amount of charge to restore the original charge density = Original charge - New charge

Now let's calculate the values step by step:

Given:
Original charge = 10.0 nC

1. New length = 1.5 * original length
Since the length was increased by 50%, the new length is 1.5 times the original length.

2. New charge = original charge
In this scenario, no new charge is added, so the new charge remains the same as the original charge, which is 10.0 nC.

3. New charge density = New charge / New length
New charge density = 10.0 nC / (1.5 * original length)

4. Charge density change factor = New charge density / Original charge density
Charge density change factor = (10.0 nC / (1.5 * original length)) / (10.0 nC / original length)
Simplifying the expression, we get:

Charge density change factor = original length / (1.5 * original length)
The original length cancels out.

Charge density change factor = 1 / 1.5
Charge density change factor = 0.6667

So, the charge density has changed by a factor of 0.6667.

5. Amount of charge to restore the original charge density = Original charge - New charge
Amount of charge to restore the original charge density = 10.0 nC - 10.0 nC
Amount of charge to restore the original charge density = 0 nC

Therefore, no additional charge needs to be added to restore the wire's original charge density.