Question

Two point charges lie on the x axis. A charge of 6.4 μC is at the origin, and a charge of -9.5 μC is at x=10.0cm.

a) What is the net electric field at x=−4.0cm? (N/C?)

b)What is the net electric field at x=+4.0cm? (N/C?)

Answer 1

a)
`Ep=3.16*10^(7) (-i) (N)/(m)` ,at x = −4.0cm

b)
`Ep=5.97*10^(7) (+i) (N)/(m)`, at x=10.0cm

Step-by-step explanation:

To solve this problem we apply the theory of the magnetic field:

Formula to calculate the magnetic field (E) at a point P due to an electric charge(q):

`E=(k*q)/(d^(2) )` Formula (1)

K: Coulomb constant in
`(N)/(m^(2)*C^(2)  )`

q= Electric charge in Coulombs( C)

d= Distance from location of q to point P in meters (m)

Concepts that we must take into account

:

• The total electric field due to a group of charges is equal to the vector sum of the electric fields of all charges.
• The electric field lines leave a positive charge and enter a negative charge.

1μC=
`10^(-6) C`

1cm=
`10^(-2) m`

Problem development

Case a) calculation of the net electric field at x = −4.0cm

Data:

q1=6.4μC =
`6.4 *10^(-6) C`, q2=-9.5μC =
`-9.5 *10^(-6) C`

`d1=4cm=4*10^(-2)m`,
`d2=14cm=14*10^(-2)m`

`k=8.99*10^(9) (N)/(m^(2)*C^(2)  )`

Ep=E1+E2 : Total electric field at point P

We use formula 1 to calculate the electric field:

Ep=
`(k*q1)/(d1^(2) )` +
`(k*q2)/(d2^(2) )` : If we factor K and replace the data, then:

`Ep=8.99*10^(9) ((6.4*10^(-6) )/((4*10^(-2))^(2) )(-i) +(9.5*10^(-6))/((14*10^(-2))^(2)) (+i))`:if we use factor
`((10^(-6) )/(10^(-4) ) )`,then:

`Ep: 8.99*10^(9) ((10^(-6) )/(10^(-4) ) ) ((6.4)/(16) (-i) +((9.5)/(196)(i))`

`Ep=(-3.6+0.4357)*10^(7) (i) (N)/(C)`

`Ep=3.16*10^(7) (-i) (N)/(m)`

Case b) calculation of the net electric field at x = +4.0cm

Data:

q1=6.4μC =
`6.4 *10^(-6) C`, q2=-9.5μC =
`-9.5 *10^(-6) C`

`d1=4cm=4*10^(-2)m`,
`d2=6cm=6*10^(-2)m`

`k=8.99*10^(9) (N)/(m^(2)*C^(2)  )`

Ep=E1+E2 : Total electric field at point P

We use formula 1 to calculate the electric field:

Ep=
`(k*q1)/(d1^(2) )` +
`(k*q2)/(d2^(2) )` : If we factor K and replace the data, then:

`Ep=8.99*10^(9) ((6.4*10^(-6) )/((4*10^(-2))^(2) )(+i) +(9.5*10^(-6))/((6*10^(-2))^(2)) (+i))`: If we factor
`((10^(-6) )/(10^(-4) ) )`,then:

`Ep: 8.99*10^(9) ((10^(-6) )/(10^(-4) ) ) ((6.4)/(16) (+i) +((9.5)/(36)(i))`

`Ep=(+3.6+2.37)*10^(7) (i) (N)/(C)`

`Ep=5.97*10^(7) (+i) (N)/(m)`