Електричні поля – проблеми та рішення

Електричні поля – проблеми та рішення

1. Point A located at the center between two charges. Both charges have the same magnitude but opposite sign and separated by a distance of a. The magnitude of the electric field at point A is 36 N/C. If point A moved 1/2a close to one of both вантажі, what is the magnitude of the електричне поле at point A?

Відомо:

Заряд 1 (q₁) = +Q

Заряд 2 (q₂) = -Q

The distance between charge 1 and point A (r_1A) = ½ a

The distance between charge 2 and point A (r_2A) = ½ a

The magnitude of the electric field at point A (E_A) = 36 NC^-1

Бажано: The magnitude of the electric field

Рішення:

Крок 1.

The electric field produced by a charge +Q в точці A :

Test charge is positive and charges 1 is positive so that the direction of the electric field points to charge 2.

The electric charge produced by a charge -Q в точці A :

Test charge is positive and charges 2 is negative so that the direction of the electric field points to charge 2.

The resultant of the electric field at point A :

Крок 2.

If point A is moved close to charge 1 then :

The distance between charge 1 and point A (r_1A) = ¼ a

The distance between charge 2 and point A (r_2A) = ¾ a

The electric field produced by charge +Q в точці A :

Test charge is positive and charges 1 is positive so that the direction of the electric field points to charge 2.

The electric field produced by charge -Q at point A :

Test charge is positive and charges 2 is negative so that the direction of the electric field points to charge 2.

The resultant of the electric field at point A :

2. Two charges q_A = 1 μC and q_B = 4 μC are separated by a distance of 4 cm (k = 9 x 10⁹ Н м² C^-2). What is the magnitude of the electric field at the center between q_A and q_B.

Відомо:

Заряд A (q_A) = 1 μC = 1 x 10^-6 C

Заряд B (q_B) = 4 μC = 4 x 10^-6 C

k = 9 x 10⁹ Н м² C^-2

Distance between charge A and B (r_AB) = 4 см = 0.04 метра

Distance between charge A and the center point (r_A) = 0.02 метрівs

Distance between charge B and the center point (r_B) = 0.02 метрівs

Відомі: The magnitude of the electric field

Рішення:

The electric field produced by charge A at the center point :

Test charge is positive and charges A is positive so that the direction of the electric field points to charge B.

The electric field produced by charge B at the center point :

Test charge is positive and charge B is positive so that the direction of the electric field points to charge A.

The resultant of the electric field at the center point :

E_A і Е_B have the opposite direction.

E = E_B - Е_A = 9 х 10^{7 -} 2.25 х 10⁷ = 6.75 х 10⁷ NC^-1

3. According to figure below, where the point P is located so that the magnitude of the electric field at point P = 0 ? (k = 9 x 10⁹ Nm²C^-2, 1 μC = 10^-6 C)

Рішення

If point P located at the left of Q₁; the electric field produced by Q₁ on point P points to leftward (від Q₁) and the electric field produced by Q₂ on point P points to rightward (point to Q₁). The direction of the electric field is opposite so that the electric field at point P = 0.

Відомо:

Q₁ = +9 μC = +9 x 10^-6 C

Q₂ = -4 μC = -4 x 10^-6 C

k = 9 x 10⁹ Nm²C^-2

Distance between charge 1 and charge 2 = 3 cm

Відстань між Q₁ and point P (r_1P) = а

Відстань між Q₂ і точка P (r_2P) = 3 + a

Розшукується: Position of point P

Рішення:

Point P located at leftward of Q₁.

The electric field produced by Q₁ в точці P :

Test charge is positive and Q₁ is positive so that the direction of the electric field to leftward.

The electric field produced by Q₂ в точці P :

Test charge is positive and Q₂ is negative so that the direction of the electric field to rightward.

Resultant of the electric field at point A :

Use quadratic formula to find a :

Відстань між Q₂ і точка P (r_2P) = 3 + a = 3 – 1.8 = 1.2 cm or 3 + a = 3 – 9 = -6 cm.

Відстань між Q₁ і точка P (r_1P) = a = -9 cm or -1.8 cm.

Point P located at 1.2 см rightward of Q₂.

4. Charge q₃ located at 5 cm rightward of q₂, as shown in the figure below. What is the magnitude of the electric field at charge q₃ (1 µC = 10^-6 С).

Рішення:

Charge q₃ is positive so that the direction of the electric field at charge q₃ points to the minus charge q₂ (E₂) and away from the plus charge q₁ (E₁). The resultant of the electric field is the sum of the electric field E₁ і Е₂.

Відомо:

Charge q₁ = 5 µC = 5 x 10^-6 Кулон

Charge q₂ = 5 µC = -5 x 10^-6 Кулон

Distance between charge q₁ and charge q₃ (r₁) = 15 см = 0.15 м = 15 x 10^-2 метри

Distance between charge q₂ and charge q₃ (r₂) = 5 см = 0.05 м = 5 x 10^-2 метри

k = 9 x 10⁹ Н м² C^-2

Розшукується: The electric field at charge q₃

Рішення:

The electric field 1 :

E₁ = k q₁ / r₁²

E₁ = (9 x 10⁹)(5 x 10^-6) / (15 x 10^-2)²

E₁ = (45 x 10³) / (225 x 10^-4)

E₁ = 0.2 х 10⁷ Н / З

The electric field 2 :

E₂ = k q₂ / r₂²

E₂ = (9 x 10⁹)(5 x 10^-6) / (5 x 10^-2)²

E₂ = (45 x 10³) / (25 x 10^-4)

E₂ = 1.8 х 10⁷ Н / З

Resultant of the electric field :

The resultant of the electric field at charge q₃ :

E = E₂ - Е₁ = (1.8 x 10⁷) – (0.2 x 10⁷) = 1.6 x 10⁷ Н / З

The direction of the electric field points to leftward (same direction as E₂).

5. Two charges are separated as shown in figure below. What is the electric field at point P (k = 9 x 10⁹ Н м² C^-2)

Рішення

Відомо:

Charge q_A = +2.5 C

Charge q_B = -2 C

Distance between charge q_A and point P (r_A) = 5 м

Distance between charge q_B and point P (r_B) = 2 м

k = 9 x 10⁹ Н м² C^-2

Розшукується: the magnitude of the electric field at point P.

Рішення:

The electric field A :

E_A = k q_A / r_A²

E_A = (9 x 10⁹)(2.5) / (5)²

E_A = (22.5 x 10⁹) / 25

E_A = 0.9 х 10⁹ Н / З

The electric field B :

E_B = k q_B / r_B²

E_B = (9 x 10⁹)(2) / (2)²

E_B = (18 x 10⁹) / 4

E_B = 4.5 х 10⁹ Н / З

Resultant of the electric field :

Resultant of the electric field at point P :

E = E_B – EA = (4.5 – 0.9) x 10⁹ = 3.6 х 10⁹ Н / З

The direction to leftward (same direction as E_B).

6. Two charges Q₁ = -40 µC and Q₂ = +5 µC as shown in figure below (k = 9 x 10⁹ Нм².C^-2 and 1 µC = 10^-6 C),. What is the magnitude of the electric field at point P.

Відомо:

Charge q₁ = -40 µC = -40 x 10^-6 C

Charge q₂ = +5 µC = +5 x 10^-6 C

Distance between q₁ and point P (r₁) = 40 см = 0.4 м = 4 x 10^-1 m

Distance between q₂ and point P (r₂) = 10 cm = 0.1 = 1 x 10^-1 m

k = 9 x 10⁹ Н м² C^-2

Розшукується: the magnitude of the electric field at point P.

Рішення:

The electric field 1 :

E₁ = k q₁ / r₁²

E₁ = (9 x 10⁹)(40 x 10^-6) / (4 x 10^-1)²

E₁ = (360 x 10³) / (16 x 10^-2)

E₁ = 22.5 х 10⁵ Н / З

The electric field 2 :

E₂ = k q₂ / r₂²

E₂ = (9 x 10⁹)(5 x 10^-6) / (1 x 10^-1)²

E₂ = (45 x 10³) / 1 x 10^-2

E₂ = 45 х 10⁵ Н / З

Resultant of the electric field :

The resultant of the electric field at point P :

E = E₂ – E1 = (45 – 22.5) x 10⁵ = 22.5 х 10⁵ Н / З

Е = 2.25 х 10⁶ Н / З

The direction of the electric field points to rightward (same direction as E₂).

7. Two point charges as shown in figure below.

Where is point P located so that the magnitude of the electric field at point P = 0. k = 9.10⁹ Nm².C^-2, 1 µC = 10^-6 C.

Відомо:

Заряд 1 (q₁) = -9 µC = -9.10^-6 Кулон

Заряд 2 (q₂) = 1 µC = 1.10^-6 Кулон

Distance between q₁ and q₂ (r₁₂) = 1см

k = 9.10⁹ Nm².C^-2

Розшукується: Position of point P

Рішення:

E₁ = the magnitude of the electric field produced by q₁ at point P

Напрямок E₁ до q₁ оскільки q₁ є негативним.

E₂ = the magnitude of the electric field produced by q₂ at point P

Напрямок E₂ від q₂ оскільки q₂ позитивний.

The electric field at point = 0.

Використайте формулу квадратного урівняння:

Distance between P and q₂ = x = 0.5 cm.

Point P located at 0.5 cm rightward q₂ or 0.25 cm leftward q₁.

8. According to the figure below, if the magnitude of the electric field at point P = k Q/x², then x = ….

Відомо:

E_P = k Q / x²

Розшукується: x

Рішення:

E₂ = The magnitude of the electric field at point P by charge +32Q

r₂ =Distance between charge +32Q and point P = a + x

Використайте формулу квадратного урівняння:

1. What is an electric field?
   • Відповідь: An electric field is a region around a charged object where electric forces can be experienced by other charged objects. It is a vector field, meaning it has both magnitude and direction at every point.
2. How is the strength of an electric field determined?
   • Відповідь: The strength or magnitude of an electric field $E$ at a point is defined as the force $F$ experienced by a positive test charge q₀​ placed at that point, divided by the magnitude of the test charge itself: E=​F/q₀​.
3. How does the electric field due to a point charge vary with distance?
   • Відповідь: The electric field $E$ due to a point charge $Q$ is inversely proportional to the square of the distance $r$ from the charge. The relationship is given by E=kQ​/r², Де $k$ is Coulomb’s constant.
4. What is the direction of the electric field due to a positive charge?
   • Відповідь: The electric field due to a positive charge points radially outward from the charge. For a negative charge, the field points radially inward, towards the charge.
5. How can you represent electric fields graphically?
   • Відповідь: Electric fields can be represented graphically using field lines (or lines of force). The direction of the field at any point is tangent to the field line at that point, and the density of the lines indicates the magnitude of the field.
6. What happens to the electric field inside a conductor in electrostatic equilibrium?
   • Відповідь: Inside a conductor in electrostatic equilibrium, the electric field is zero. This is because any external field causes free electrons in the conductor to redistribute, cancelling the external field inside.
7. How do electric field lines behave near a sharp edge of a conductor?
   • Відповідь: Near a sharp edge or pointed tip of a conductor, the electric field lines are more concentrated, leading to a stronger electric field in that region. This is the basis for the operation of devices like the lightning rod.
8. How do superposition principles apply to electric fields?
   • Відповідь: The electric field due to multiple charges at a point is simply the vector sum of the electric fields due to each individual charge. This is known as the superposition principle.
9. How is the work done by an external agent related to the electric field when moving a charge within the field?
   • Відповідь: The work $W$ done by an external agent in moving a charge $q$ from one point to another in an electric field is equal to the negative of the change in electric potential energy, which is $W=q\times \Delta V$, Де $\Delta V$ is the change in electric potential.

10. Can electric field lines ever cross each other?

    • Відповідь: No, electric field lines cannot cross each other. If they did, it would imply that at the point of intersection, there are two different directions of the electric field, which is not possible.